Discrete Mathematics I: Foundations of Logic
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Questions and Answers

Which of the following is the contrapositive of the statement 'If today is Friday, then I have a test today'?

  • If I have a test today, then today is Friday.
  • If I don’t have a test today, then today is Friday.
  • If today is not Friday, then I have a test today.
  • If today isn’t Friday, then I don’t have a test today. (correct)
  • The inverse of the statement 'The home team wins whenever it is raining' is 'If it is not raining, then the home team wins.'

    False (B)

    What is a Boolean variable?

    A variable that can be either true or false.

    In propositional logic, a bit represents _____ and _____ values.

    <p>0, 1</p> Signup and view all the answers

    What is the converse of the statement 'If the home team wins, then it is raining'?

    <p>If the home team wins, then it is raining. (B)</p> Signup and view all the answers

    Match the logical operator with its description:

    <p>∨ = Logical OR ∧ = Logical AND ¬ = Logical NOT ⊕ = Logical XOR</p> Signup and view all the answers

    What is the symbol used to denote 'negation'?

    <p>¬p (A)</p> Signup and view all the answers

    The statement 'Julienne’s smartphone has at least 32GB of memory' can be negated as 'Julienne’s smartphone has less than 32GB of memory'.

    <p>True (A)</p> Signup and view all the answers

    A truth table can be constructed for the compound proposition (p ∨ q) → (p ⊕ q).

    <p>True (A)</p> Signup and view all the answers

    What is represented by 1 and 0 in Boolean logic?

    <p>1 represents true, and 0 represents false.</p> Signup and view all the answers

    What does the symbol 'XOR' stand for?

    <p>exclusive or</p> Signup and view all the answers

    The conjunction of two propositions, p and q, is denoted by __________.

    <p>p ∧ q</p> Signup and view all the answers

    Match the following logical operations with their definitions:

    <p>Conjunction = p and q are both true Disjunction = At least one of p or q is true Negation = It is not the case that p Biconditional = p is true if and only if q is true</p> Signup and view all the answers

    Which of the following statements is the negation of 'There is no pollution in Iloilo'?

    <p>There is pollution in Iloilo. (D)</p> Signup and view all the answers

    The biconditional operation is denoted by p ↔ q.

    <p>True (A)</p> Signup and view all the answers

    What is the truth value of p ∧ q when both p and q are true?

    <p>true</p> Signup and view all the answers

    What is the truth value of $x ∧ y$ when both $x$ and $y$ are true?

    <p>True (A)</p> Signup and view all the answers

    ~$y ∧ x$ is true when $x$ is false and $y$ is true.

    <p>False (B)</p> Signup and view all the answers

    What is the result of the conjunction ~x ∧ y when $x$ is true and $y$ is true?

    <p>False</p> Signup and view all the answers

    In the conjunction $x ∧ y$, if $x$ is false and $y$ is true, the result is _____.

    <p>False</p> Signup and view all the answers

    Match the following expressions with their expected outcomes:

    <p>$x ∧ y$ = False when either $x$ or $y$ is false ~$x ∧ y$ = True when $x$ is false and $y$ is true ~$y ∧ x$ = True when $y$ is false and $x$ is true</p> Signup and view all the answers

    Which statement correctly describes the outcome of ~x ∧ y when $x$ is False and $y$ is False?

    <p>False (B)</p> Signup and view all the answers

    The truth value of ~x ∧ y will always be true if $y$ is true.

    <p>False (B)</p> Signup and view all the answers

    List the possible outcomes for the conjunction $x ∧ y$ when both $x$ and $y$ are false.

    <p>False</p> Signup and view all the answers

    If $y$ is true and $x$ is false, the result of the expression $~y ∧ x$ is _____.

    <p>False</p> Signup and view all the answers

    What is the origin of the term 'algorithm'?

    <p>A mathematician's name (A)</p> Signup and view all the answers

    An algorithm should produce an output after an infinite number of steps.

    <p>False (B)</p> Signup and view all the answers

    What property of an algorithm ensures that it produces correct output values for each set of input values?

    <p>Correctness</p> Signup and view all the answers

    A[n] __________ algorithm selects the best choice at each step instead of considering all sequences.

    <p>greedy</p> Signup and view all the answers

    Match the properties of algorithms with their definitions:

    <p>Input = Values from a specified set Output = Solution values from input Effectiveness = Performable steps in finite time Generality = Applicable for all desired problem forms</p> Signup and view all the answers

    Which of the following is NOT a property of algorithms?

    <p>Infinity (C)</p> Signup and view all the answers

    The steps of an algorithm must be defined precisely.

    <p>True (A)</p> Signup and view all the answers

    What does 'finiteness' mean in the context of algorithms?

    <p>It means an algorithm should produce desired output after a finite number of steps.</p> Signup and view all the answers

    What does a compound proposition need to be considered satisfiable?

    <p>There must be at least one assignment of truth values that makes it true. (C)</p> Signup and view all the answers

    A compound proposition is unsatisfiable if its negation is a tautology.

    <p>True (A)</p> Signup and view all the answers

    What is the result of applying De Morgan's law to the expression ¬ ¬p ∧ q?

    <p>p ∨ ¬q ∧ (p ∨ q)</p> Signup and view all the answers

    To show that a compound proposition is unsatisfiable, every assignment of truth values must make it _____ .

    <p>false</p> Signup and view all the answers

    Match the following terms with their definitions:

    <p>Satisfiable = Has at least one assignment that makes it true Unsatisfiable = False for all assignments of truth values Tautology = True for all assignments of truth values Proposition = A statement that can be either true or false</p> Signup and view all the answers

    Which of the following statements correctly describes the use of quantifiers in first-order logic?

    <p>Quantifiers help define the range of variables in logical expressions. (D)</p> Signup and view all the answers

    The expression ¬(p ∧ q) is logically equivalent to ¬p ∨ ¬q.

    <p>True (A)</p> Signup and view all the answers

    What is the result of applying the identity law for F to the expression p ∨ F?

    <p>p</p> Signup and view all the answers

    What is the purpose of establishing that f(x) is O(g(x))?

    <p>To demonstrate that f(x) grows slower than g(x) (A), To establish a relationship between the two functions (B)</p> Signup and view all the answers

    There can be only one pair of witnesses for the big-O relationship between f(x) and g(x).

    <p>False (B)</p> Signup and view all the answers

    What values need to be found to prove that f(x) is O(g(x))?

    <p>Constants C and k</p> Signup and view all the answers

    Paul Bachman introduced the big-O notation in the year _____

    <p>1892</p> Signup and view all the answers

    Match the following big-O concepts with their definitions:

    <p>C = The constant in the big-O notation k = The threshold for x in the big-O notation f(x) = The function being analyzed g(x) = The function used as the comparison base</p> Signup and view all the answers

    When proving f(x) = x^2 + 2x + 1 is O(x^2), which value of k is used?

    <p>1 (C)</p> Signup and view all the answers

    The function f(x) = 7x^2 is also O(x^3).

    <p>True (A)</p> Signup and view all the answers

    What is the result of replacing x and 1 in the function f(x) = x^2 + 2x + 1?

    <p>4x^2</p> Signup and view all the answers

    What is the truth value of the statement 'If $x^2 \geq 0$ then $x \geq 0$'?

    <p>False (B)</p> Signup and view all the answers

    The contrapositive of a statement always has the same truth value as the original statement.

    <p>True (A)</p> Signup and view all the answers

    What is the definition of the inverse in logic?

    <p>The inverse is formed by negating both the hypothesis and the conclusion of a conditional statement.</p> Signup and view all the answers

    The statement 'If it rains, then the ground is wet' can be reversed to form its converse as 'If the ground is wet, then _____'.

    <p>it rains</p> Signup and view all the answers

    What is the logical representation of the statement 'You do not have Sprite'?

    <p>¬p (C)</p> Signup and view all the answers

    The biconditional statement 'p if and only if q' is true if both p and q have the same truth value.

    <p>True (A)</p> Signup and view all the answers

    Provide an example of a statement and its contrapositive.

    <p>'If A, then B' and 'If not B, then not A'.</p> Signup and view all the answers

    Which statement correctly describes the implication of $p → q$?

    <p>If p then q. (B)</p> Signup and view all the answers

    The biconditional statement $p ↔ q$ is false when p and q have different truth values.

    <p>True (A)</p> Signup and view all the answers

    What is the definition of a biconditional statement?

    <p>A biconditional statement is 'p if and only if q'.</p> Signup and view all the answers

    The statement 'If -1 is a positive number, then 2 + 2 = 5' is considered _____ due to the false premise.

    <p>true</p> Signup and view all the answers

    Match each implication with its equivalent phrase:

    <p>p → q = p implies q p ∧ q = p and q p ↔ q = p if and only if q ¬p = not p</p> Signup and view all the answers

    Which of the following is NOT true regarding implications?

    <p>If both p and q are false, the implication is false. (C)</p> Signup and view all the answers

    The conclusion of the implication 'If sin x = 0, then x = 0' is valid for all values of x.

    <p>False (B)</p> Signup and view all the answers

    In the statement 'p is a sufficient condition for q', p implies that _____ must be true.

    <p>q</p> Signup and view all the answers

    What is a bound variable in the context of quantifiers?

    <p>A variable that is associated with a quantifier. (B)</p> Signup and view all the answers

    The expression ∃x P(x) indicates that there exists at least one x such that P(x) is true.

    <p>True (A)</p> Signup and view all the answers

    What is the scope of a quantifier?

    <p>The formula immediately following the quantifier.</p> Signup and view all the answers

    In the expression ∀x P(x), the variable x is considered ______.

    <p>bound</p> Signup and view all the answers

    Match the expression to its quantifier:

    <p>∀x P(x) = For all x ∃x Q(x) = There exists x ∀y R(y) = For every y ∃z S(z) = Some z exists</p> Signup and view all the answers

    What does the expression ∀x P(x) Q(x) signify?

    <p>P(x) is true for all values of x. (B)</p> Signup and view all the answers

    A free variable in a predicate can be replaced with any value without affecting the truth value of the predicate.

    <p>True (A)</p> Signup and view all the answers

    How would you symbolize 'Some integers are multiples of 7'?

    <p>∃x P(x) Q(x)</p> Signup and view all the answers

    What characterizes a greedy algorithm?

    <p>It picks the best option available at each step (D)</p> Signup and view all the answers

    A greedy algorithm is guaranteed to find an optimal solution.

    <p>False (B)</p> Signup and view all the answers

    What is the purpose of asymptotic analysis?

    <p>To compute the running time of an algorithm in mathematical units of computation.</p> Signup and view all the answers

    The rate of growth of a function can be described through __________ analysis.

    <p>asymptotic</p> Signup and view all the answers

    Match the following functions with their characteristics:

    <p>fA(n) = 30n + 8 = Linear growth fB(n) = n^2 + 1 = Quadratic growth f(n) = log(n) = Logarithmic growth f(n) = 2^n = Exponential growth</p> Signup and view all the answers

    Given Program A takes $f_A(n)=30n+8$ microseconds, and Program B takes $f_B(n)=n^2+1$ microseconds, which program is more efficient for large values of n?

    <p>Program A (A)</p> Signup and view all the answers

    If $f(x)$ is faster growing than $g(x)$, then for large enough values of x, $f(x)$ will always be smaller than $g(x)$.

    <p>False (B)</p> Signup and view all the answers

    What does it mean when we say a greedy algorithm has found a feasible solution?

    <p>It means the solution meets all the necessary constraints but may not be the best possible.</p> Signup and view all the answers

    Which statement represents the contrapositive of the implication 'If today is Friday, then I have a test today'?

    <p>If today isn't Friday, then I don't have a test today. (D)</p> Signup and view all the answers

    The inverse of the statement 'The home team wins whenever it is raining' is 'If it is not raining, then the home team does not win.'

    <p>True (A)</p> Signup and view all the answers

    What logical operator is used to denote 'and'?

    <p>∧</p> Signup and view all the answers

    In Boolean logic, the value _______ represents true.

    <p>1</p> Signup and view all the answers

    What is the truth value of the expression $p ∨ q$ when both p and q are false?

    <p>False (B)</p> Signup and view all the answers

    A biconditional statement 'p ↔ q' is true when both p and q have the same truth value.

    <p>True (A)</p> Signup and view all the answers

    What is the purpose of constructing a truth table?

    <p>To determine the truth values of compound propositions.</p> Signup and view all the answers

    What is the result of the disjunction 𝑝 ∨ 𝑞 when both 𝑝 and 𝑞 are false?

    <p>False (B)</p> Signup and view all the answers

    The exclusive or (𝑝 ⊕ 𝑞) is true when both 𝑝 and 𝑞 are true.

    <p>False (B)</p> Signup and view all the answers

    What is the negation of the proposition '𝑎 is true'?

    <p>~𝑎</p> Signup and view all the answers

    The disjunction of 𝑎 and 𝑏 is denoted as ______.

    <p>𝑎 ∨ 𝑏</p> Signup and view all the answers

    Match the following logical operations with their truth table outcomes:

    <p>𝑎 ∨ 𝑏 = True if at least one is true 𝑎 ∨ ~𝑏 = True if a is true or b is false ~𝑎 ∨ 𝑏 = True if not a is true or b is true 𝑝 ⊕ 𝑞 = True if exactly one of p or q is true</p> Signup and view all the answers

    Which of the following correctly describes the truth table for 𝑎 ∨ ~𝑏?

    <p>True if a is true, regardless of b (C)</p> Signup and view all the answers

    The truth table for exclusive or (𝑝 ⊕ 𝑞) results in true when both propositions are false.

    <p>False (B)</p> Signup and view all the answers

    In the truth table for disjunction, the only case in which 𝑝 ∨ 𝑞 is false is when both 𝑝 and 𝑞 are ______.

    <p>false</p> Signup and view all the answers

    When is a conditional statement p → q considered false?

    <p>When p is true and q is false (D)</p> Signup and view all the answers

    The statement 'If Maria learns discrete mathematics, then she will find a good job' can be represented as p → q.

    <p>True (A)</p> Signup and view all the answers

    What does the XOR operator signify in logical operations?

    <p>It signifies 'exclusive or', meaning one condition is true, but not both.</p> Signup and view all the answers

    In a truth table for the conditional statement $p → q$, if $p$ is false, the truth value of $p → q$ is _____ regardless of $q$.

    <p>true</p> Signup and view all the answers

    Which of the following statements represents a conditional statement?

    <p>If it rains, then the ground will be wet. (C)</p> Signup and view all the answers

    The statement 'Maria will find a good job unless she does not learn discrete mathematics' is equivalent to the conditional statement p → q.

    <p>False (B)</p> Signup and view all the answers

    The symbol that represents a conditional statement is _____ .

    <p>→</p> Signup and view all the answers

    What type of proof directly uses axioms, definitions, and earlier theorems to establish a conclusion?

    <p>Direct Proof (A)</p> Signup and view all the answers

    In an indirect proof, if not q then not p is inferred from the premise.

    <p>True (A)</p> Signup and view all the answers

    What is the conclusion established in the direct proof example regarding odd integers?

    <p>If x is odd, then x^2 is odd.</p> Signup and view all the answers

    A proof by __________ shows that if some statement were true, a logical contradiction occurs, hence the statement must be false.

    <p>contradiction</p> Signup and view all the answers

    Match the following proof techniques with their descriptions:

    <p>Direct Proof = Establishes conclusion using axioms and earlier theorems Indirect Proof = Infers conclusion from the contrapositive Proof by Contradiction = Demonstrates that assuming a statement leads to a contradiction Proof by Exhaustion = Considers all possible cases to establish a conclusion</p> Signup and view all the answers

    Which of the following statements accurately expresses the meaning of the implication $p \to q$?

    <p>p only if q (C)</p> Signup and view all the answers

    What is the definition of proving that if x² is even, then x must be even, using indirect proof?

    <p>If x is not even, then x² is not even. (D)</p> Signup and view all the answers

    Which of the following statements is the converse of the implication 'If today is Friday, then I have a test today'?

    <p>If I have a test today, then today is Friday. (B)</p> Signup and view all the answers

    The biconditional statement $p \leftrightarrow q$ is true when one of the statements is true and the other is false.

    <p>False (B)</p> Signup and view all the answers

    The product of two odd numbers is __________.

    <p>odd</p> Signup and view all the answers

    What is the truth value of the statement 'If sin x = 0, then x = 0'?

    <p>false</p> Signup and view all the answers

    The contrapositive of the statement 'If the home team wins, then it is raining' is 'If it is not raining, then the home team wins.'

    <p>False (B)</p> Signup and view all the answers

    What is the example used to demonstrate a proof by contradiction regarding the number 2?

    <p>Prove that 2 is irrational.</p> Signup and view all the answers

    What is the inverse of the statement 'If I have a test today, then today is Friday'?

    <p>If I don't have a test today, then today is not Friday.</p> Signup and view all the answers

    In the biconditional statement 'You can take the flight if and only if you buy a ticket,' the statement represents the logical form of ______.

    <p>p ↔ q</p> Signup and view all the answers

    Match the following logical implications with their meanings:

    <p>p implies q = If p then q p only if q = q is necessary for p p is sufficient for q = If p, q holds true q follows from p = p leading to q</p> Signup and view all the answers

    In Boolean logic, a variable can take on the values of ______ or ______.

    <p>true, false</p> Signup and view all the answers

    Which of the following statements is true regarding conditional statements?

    <p>A false premise allows for any conclusion to be valid. (B)</p> Signup and view all the answers

    If the statement 'x^2 + y^2 = 0' is true, then both x and y must equal zero.

    <p>True (A)</p> Signup and view all the answers

    What does the symbol ~ represent in Boolean logic?

    <p>Negation (D)</p> Signup and view all the answers

    The logical operator 'if and only if' is represented by the symbol ______.

    <p>↔</p> Signup and view all the answers

    List the outcomes for the conjunction $x ∧ y$ when both $x$ and $y$ are false.

    <p>false</p> Signup and view all the answers

    The statement 'If $x^2 eq 0$, then $x eq 0$' is an example of an implication.

    <p>True (A)</p> Signup and view all the answers

    What does the symbol ∀ represent in the context of quantifiers?

    <p>For all (B)</p> Signup and view all the answers

    The statement 'All odd integers are prime' is a valid conclusion based on the logical predicate definitions given.

    <p>False (B)</p> Signup and view all the answers

    What is a theorem?

    <p>A statement that can be shown to be true under certain conditions.</p> Signup and view all the answers

    The squares of all ______ integers are odd.

    <p>odd</p> Signup and view all the answers

    Match the following logical statements with their types:

    <p>6 is an even integer. = Fact If x is even, then x + 1 is odd. = Conditional statement x is even if and only if x is divisible by 2. = Biconditional statement</p> Signup and view all the answers

    Which statement correctly reflects the meaning of the symbol ∃?

    <p>There exists at least one (D)</p> Signup and view all the answers

    For all integers x, if x is odd, then x^2 is odd.

    <p>True (A)</p> Signup and view all the answers

    Provide an example of a statement that is shown as a bi-implication.

    <p>For all integers x, x is even if and only if x is divisible by 2.</p> Signup and view all the answers

    What does the notation $f(x) = \Theta(g(x))$ indicate?

    <p>f(x) is bounded both above and below by g(x) (D)</p> Signup and view all the answers

    For the function $f(n) = 4n + 6$, it is true that $C_1 = 4$ and $C_2 = 6$.

    <p>True (A)</p> Signup and view all the answers

    Identify the value of $k$ when proving that $f(n) = 4n + 6$ is $ heta(g(n))$.

    <p>3</p> Signup and view all the answers

    In big-Theta notation, the functions must satisfy the inequality $C_1 g(n) \leq f(n) \leq C_2 g(n)$ for all _____.

    <p>n ≥ k</p> Signup and view all the answers

    Match the following functions with their corresponding asymptotic notations:

    <p>4n + 6 = $\Theta(n)$ 3x^2 + 2x + 3 = $O(x^2)$ x^2 = $\Omega(x^2)$ 2n + 5 = $O(n)$</p> Signup and view all the answers

    Given $f(n) = 4n + 6$, what is the upper bound $C_2$ for this function?

    <p>6n (A)</p> Signup and view all the answers

    The inequality $4n + 6 oundup{≤} 2n$ is a correct step in proving big-O for $f(n) = 4n + 6$.

    <p>False (B)</p> Signup and view all the answers

    Define 'Big-O notation' in relation to function growth.

    <p>Big-O notation describes an upper bound on the growth rate of a function.</p> Signup and view all the answers

    What is a key characteristic of a greedy algorithm?

    <p>It finds a feasible solution quickly. (C)</p> Signup and view all the answers

    Greedy algorithms are also known as the shortest path algorithms.

    <p>True (A)</p> Signup and view all the answers

    What is the main concept of asymptotic analysis?

    <p>It computes the running time of operations in terms of growth rates for large inputs.</p> Signup and view all the answers

    Program A takes ___ microseconds to process n records compared to Program B.

    <p>30n + 8</p> Signup and view all the answers

    Which of the following statements about logical implications is true?

    <p>Only the contrapositive has the same truth value as the original implication. (D)</p> Signup and view all the answers

    Match the following growth rates with their descriptions:

    <p>fA(n) = Linear growth fB(n) = Quadratic growth</p> Signup and view all the answers

    If f(x) is faster growing than g(x), what does this mean?

    <p>f(x) will eventually become larger than g(x) as x grows. (A)</p> Signup and view all the answers

    The statement 'If the moon is made of cheese, then I am an astronaut' has a true inverse.

    <p>False (B)</p> Signup and view all the answers

    What is the result of the conjunction $p ∧ q$ when both $p$ and $q$ are true?

    <p>true</p> Signup and view all the answers

    It is sufficient to show only one counterexample to prove that a greedy algorithm is optimal.

    <p>False (B)</p> Signup and view all the answers

    The statement 'If it is snowing, then it is cold' can be negated as '__________'.

    <p>It is snowing and it is not cold</p> Signup and view all the answers

    What would be an appropriate choice for processing millions of user records based on the given examples?

    <p>Program A</p> Signup and view all the answers

    Match the terms with their definitions:

    <p>Truth Table = A table that shows truth values for logical expressions Converse = The statement formed by switching the hypothesis and conclusion Inverse = The statement formed by negating both the hypothesis and conclusion Contrapositive = The statement formed by negating and switching the hypothesis and conclusion</p> Signup and view all the answers

    When both statements $p$ and $q$ are false, what is the truth value of $p ∧ q$?

    <p>False (B)</p> Signup and view all the answers

    What is the contrapositive of the statement 'If today is Friday, then I have a test today'?

    <p>If today isn't Friday, then I don't have a test today. (C)</p> Signup and view all the answers

    What does the statement $p ⊕ q$ represent?

    <p>Exclusively true when either $p$ or $q$ is true, but not both.</p> Signup and view all the answers

    Construct the converse for the statement: 'If it is raining, then the home team wins'.

    <p>If the home team wins, then it is raining.</p> Signup and view all the answers

    The statement 'If $2 + 2 = 4$, then $2 < 2$' is true.

    <p>False (B)</p> Signup and view all the answers

    'If I don't have a test today, then ______ is not Friday.'

    <p>today</p> Signup and view all the answers

    Match the logical operators with their definitions:

    <p>∨ = Logical disjunction (OR) ∧ = Logical conjunction (AND) ¬ = Logical negation (NOT) ⊕ = Exclusive OR (XOR)</p> Signup and view all the answers

    Which of the following correctly describes a Boolean variable?

    <p>A variable that can be true or false. (D)</p> Signup and view all the answers

    If $p$ is true and $q$ is false, then the statement $p ∧ q$ is true.

    <p>False (B)</p> Signup and view all the answers

    The symbol for logical implication is ______.

    <p>→</p> Signup and view all the answers

    Which of the following is a condition for a function f(x) to be $ heta(g(x))$?

    <p>f(x) must be O(g(x)) and Ω(g(x)) (B)</p> Signup and view all the answers

    The notation $f(n)$ is $ heta(4n + 6)$ is true for all n ≥ 3.

    <p>True (A)</p> Signup and view all the answers

    What are the values of C1 and C2 when determining big-Theta for the function f(n) = 4n + 6?

    <p>C1 = 4, C2 = 6</p> Signup and view all the answers

    If $f(x)$ is $ heta(g(x))$, then $C_1 g(x) ext{ ≤ } f(x) ext{ ≤ } C_2 g(x)$ where C1 = _____ and C2 = _____.

    <p>lower bound, upper bound</p> Signup and view all the answers

    Match the following expressions with their meanings:

    <p>O Notation = Upper bound growth rate Ω Notation = Lower bound growth rate θ Notation = Tight bound growth rate C1 and C2 = Constants involved in bounds</p> Signup and view all the answers

    Which of the following represents a condition for a compound proposition to be satisfiable?

    <p>There exists an assignment of truth values that makes it true. (A)</p> Signup and view all the answers

    A compound proposition is unsatisfiable if its negation is true for at least one assignment of truth values.

    <p>False (B)</p> Signup and view all the answers

    What is the purpose of finding k when proving that f(n) is O(g(n))?

    <p>To specify a point from which the inequality holds (C)</p> Signup and view all the answers

    What operation does the symbol ⊕ represent in propositional logic?

    <p>Bitwise XOR (D)</p> Signup and view all the answers

    What is the result of applying double negation to the proposition ¬¬p?

    <p>p</p> Signup and view all the answers

    For the function $f(n) = 4n + 6$, C2 can be any value greater than 6.

    <p>False (B)</p> Signup and view all the answers

    The bitwise AND operation of two bit strings always results in a string of the same length.

    <p>True (A)</p> Signup and view all the answers

    What is the relationship indicated by Big-Theta notation?

    <p>Tight bound between two functions</p> Signup and view all the answers

    A statement that is true for all assignments of truth values is known as a __________.

    <p>tautology</p> Signup and view all the answers

    Match the logical terms with their definitions:

    <p>Satisfiable = True for at least one assignment of truth values Unsatisfiable = False for all assignments of truth values Tautology = True for all assignments of truth values Solution = A specific assignment that makes a proposition true</p> Signup and view all the answers

    What is the length of the bit string '1100'?

    <p>4</p> Signup and view all the answers

    The bitwise _______ operation returns a string where each bit is 1 if at least one of the corresponding bits of the input strings is 1.

    <p>OR</p> Signup and view all the answers

    Which law is applied to simplify ¬¬p ∧ q ∧ (p ∨ q) to p ∨ (¬q ∧ q)?

    <p>Distributive (B)</p> Signup and view all the answers

    The expression ¬q ∧ q is always true.

    <p>False (B)</p> Signup and view all the answers

    Match the following bitwise operations with their outcomes:

    <p>Bitwise OR = 1 if at least one bit is 1 Bitwise AND = 1 if both bits are 1 Bitwise XOR = 1 if the bits are different Bitwise NOT = Inverts the bits</p> Signup and view all the answers

    To prove that a compound proposition is unsatisfiable, one must demonstrate that it is false for every assignment of __________.

    <p>truth values</p> Signup and view all the answers

    What is the result of the bitwise XOR operation on the bit strings '1010' and '1100'?

    <p>0110 (B)</p> Signup and view all the answers

    Inconsistent system specifications can lead to successful software development.

    <p>False (B)</p> Signup and view all the answers

    What applications can propositional logic be used for in computer science?

    <p>Designing circuits, software specifications, program verification, expert systems.</p> Signup and view all the answers

    Flashcards

    Negation of a Proposition

    The opposite truth value of a proposition.

    Logical Disjunction (OR)

    A logical operator that is true if at least one of the propositions is true.

    Exclusive OR (XOR)

    A logical operator that is true if exactly one of the propositions is true.

    Conditional (Implication)

    A logical operator that is false only when the antecedent is true and the consequent is false.

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    Biconditional

    A logical operator that is true if both propositions have the same truth value.

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    Conjunction (AND)

    A logical operator that is true only if both propositions are true.

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    Truth Table

    A table that shows all possible truth values for a logical operation.

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    Proposition

    A statement that is either true or false.

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    Conjunction (𝑥 ∧ 𝑦)

    A logical operator that is true only if both propositions (𝑥 and 𝑦) are true.

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    ~𝑥 ∧ 𝑦

    Conjunction of proposition (𝑥) negated and proposition (𝑦).

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    ~𝑦 ∧ 𝑥

    Conjunction of proposition (𝑦) negated and proposition (𝑥).

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    Logical Operator

    Connects propositions to create more complex statements.

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    Disjunction

    Logical connective. A statement that’s true if at least one of the propositions is true.

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    Converse

    An implication formed by swapping the hypothesis and conclusion of the original implication.

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    Inverse

    An implication formed by negating both the hypothesis and the conclusion of the original implication.

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    Contrapositive

    An implication formed by swapping the hypothesis and conclusion of the original implication and negating both.

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    What is a bit?

    A binary digit, representing 1 or 0, representing true or false respectively.

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    Boolean Variable

    A variable that can only have two values: true or false.

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    Algorithm

    A set of well-defined instructions for solving a problem in a finite number of steps.

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    Input (Algorithm)

    The values an algorithm receives to start its process.

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    Logical Equivalence

    Two propositions are logically equivalent if they have the same truth value for all possible truth assignments.

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    Output (Algorithm)

    The solution or result produced by the algorithm.

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    Precedence of Operators

    The order in which logical operators are evaluated in a complex proposition.

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    Definiteness (Algorithm)

    Each step in an algorithm must be clear and unambiguous.

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    Correctness (Algorithm)

    An algorithm always produces the correct output for any valid input.

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    Finiteness (Algorithm)

    An algorithm must terminate after a finite number of steps.

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    Effectiveness (Algorithm)

    Each step in an algorithm must be feasible and performable within a reasonable time.

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    Generality (Algorithm)

    An algorithm should be applicable to all problems of a similar type.

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    Satisfiable Proposition

    A proposition that can be made true by assigning truth values to its variables.

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    Unsatisfiable Proposition

    A proposition that cannot be made true by any assignment of truth values to its variables.

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    Tautology

    A proposition that is true for all possible truth value assignments.

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    Solution of a Satisfiability Problem

    An assignment of truth values to variables that makes a satisfiable proposition true.

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    Quantifier

    A symbol that specifies the quantity of elements in a set that satisfy a given condition.

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    First-Order Logic

    Part of logic that uses quantifiers to express statements about objects and properties.

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    Predicate

    A statement that becomes a proposition when variables are replaced with specific values.

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    Propositional Satisfiability

    The problem of determining whether a propositional formula can be made true by assigning truth values to its variables.

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    Big-O Notation

    A mathematical notation used to describe the limiting behavior of a function, especially as the argument tends towards infinity. It is used in computer science to classify the efficiency of algorithms and data structures.

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    Witnesses (Big-O)

    In Big-O notation, witnesses are values (C and k) that satisfy the inequality |f(x)| ≤ C|g(x)| for all x > k. They prove that f(x) is O(g(x)).

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    Finding Witnesses

    To find witnesses (C and k) for Big-O notation, first select a value of k for which the size of |f(x)| can be readily estimated when x > k. Then, use this estimate to find a value for C that satisfies the inequality |f(x)| ≤ C|g(x)| for all x > k.

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    Multiple Witness Pairs

    If one pair of witnesses exists to prove that f(x) is O(g(x)), there are infinitely many pairs of witnesses.

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    Example: Witness Proof

    To prove that f(x) = x² + 2x + 1 is O(x²), we can choose k = 1 and C = 4. This satisfies the inequality |f(x)| ≤ C|g(x)|, making 1 and 4 witnesses to this relationship.

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    Big-O Not Unique

    A function can be O(g(x)) for different functions g(x). For example, f(x) = x² + 2x + 1 is O(x²) and also O(x³).

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    Meaning of Big-O

    Big-O notation helps to understand the growth rate of a function in relation to another function. It allows us to compare and analyze the efficiency of algorithms and data structures.

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    Importance of Big-O

    Big-O notation is crucial in computer science for analyzing and comparing the efficiency of algorithms and data structures. It helps to predict their performance for large inputs, enabling developers to choose the most efficient solutions.

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    Disjunction (OR)

    A logical operator denoted by '∨' that is true if at least one of the propositions is true, and false only when both propositions are false.

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    Constructing Truth Tables

    The process of creating a table that displays the truth values of a logical expression for every possible combination of truth values of the input propositions.

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    Negation (~)

    A logical operator that reverses the truth value of a proposition. If the proposition is true, the negation is false, and vice versa.

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    Symbolic Representation

    Using symbols like '∨', '⊕', and '~' to represent logical operators in propositions.

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    Conditional Statement

    A statement that asserts if one proposition (antecedent) is true, then another proposition (consequent) is also true. It tests the relationship between two propositions, and only proves false when the antecedent is true while the consequent is false.

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    Truth Table for Conditional Statement

    A table displaying all possible truth values for a conditional statement based on the truth values of the antecedent and consequent. It helps visualize how the truth of the statement depends on the truth of its parts.

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    Biconditional Statement

    A statement asserting that two propositions are equivalent. It is true only when both propositions have the same truth value, and false otherwise.

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    Truth Table for Biconditional Statement

    A table showing all possible truth values for a biconditional statement based on the truth values of both propositions. It helps visualize when this if and only if statement is true or false.

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    Sufficient Condition in Implication

    The antecedent of a conditional statement, which, if true, guarantees the truth of the consequent. It's enough (sufficient) to make the conclusion hold.

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    Necessary Condition in Implication

    The consequent of a conditional statement, which must be true for the antecedent to be true. It's required (necessary) for the condition to hold.

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    Which of the following bicondtionals is true?

    This is a question that assesses understanding of biconditional statements and how to determine their truth value. The prompt asks the student to analyze a given biconditional statement and determine if it is true or false.

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    What is a proposition?

    A statement that can be either true or false, but not both. It's a declarative sentence that makes an assertion about reality.

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    Converse of an Implication

    An implication created by swapping the hypothesis and conclusion of the original implication.

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    Inverse of an Implication

    An implication derived by negating both the hypothesis and conclusion of the original implication.

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    Contrapositive of an Implication

    An implication created by both swapping the hypothesis and conclusion AND negating both parts of the original implication.

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    Compound Proposition

    A proposition created by combining simpler propositions using logical operators (like AND, OR, NOT).

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    If and only if

    A logical connective (symbol: ⇔) that is true only when both propositions have the same truth value, and false otherwise. It expresses a mutual implication, meaning that both propositions are either true or false together.

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    Hypothesis

    The statement that is assumed to be true in a conditional statement. It's the 'if' part of an 'if-then' statement.

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    Conclusion

    The statement that is claimed to be true if the hypothesis is true. It's the 'then' part of an 'if-then' statement.

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    Equivalent Statements

    Conditional statements that always have the same truth value. These statements can be substituted for each other without changing the logical meaning of the original statement.

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    Bound Variable

    A variable in a predicate that is directly associated with a quantifier. The quantifier dictates the scope and meaning of the variable within the expression.

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    Free Variable

    A variable in a predicate that is not bound by any quantifier. Its value can vary independently and is not restricted to specific values.

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    Quantifier Scope

    The part of a logical formula that is affected by a quantifier. It starts immediately after the quantifier and includes any variables bound by it.

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    Example 1: Square of a real number

    The statement 'the square of any real number is greater than or equal to zero' can be represented using quantifiers, predicates, and logical connectives as: ∀𝑥 P(𝑥) → Q(𝑥), where P(𝑥) represents '𝑥 is a real number' and Q(𝑥) represents '𝑥² ≥ 0'.

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    Example 2: Multiples of 7

    The statement 'some integers are multiples of 7' can be represented using quantifiers, predicates, and logical connectives as: ∃𝑥 P(𝑥) ∧ Q(𝑥), where P(𝑥) represents '𝑥 is an integer' and Q(𝑥) represents '𝑥 is a multiple of 7'.

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    Example 3: Integer squared to 16

    The statement 'there is an integer x such that x² = 16' can be represented using quantifiers and predicates as: ∃𝑥 Q(𝑥), where Q(𝑥) represents '𝑥² = 16'.

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    Greedy Algorithm

    An algorithm that makes locally optimal choices at each step, hoping to find a globally optimal solution. It may not always find the best solution, but it's efficient.

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    Optimal Solution

    The best possible solution to a problem, achieving the desired goal with the most favorable outcome.

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    Asymptotic Analysis

    A method of analyzing the running time of an algorithm for large inputs, focusing on how the time grows as the input size increases.

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    Rate of Growth

    How quickly a function's output increases as the input size grows. Used to compare the efficiency of algorithms.

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    Why Use Asymptotic Analysis?

    To understand how an algorithm's time complexity scales with larger input sizes, helping to choose the most efficient solution for large datasets.

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    Faster Growing Function

    A function that eventually becomes larger than another function for sufficiently large input values.

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    Counterexample

    A specific case that demonstrates that a general statement or algorithm is not always true.

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    What is a Counterexample In Algorithms?

    A specific input to an algorithm that shows it does not produce the expected or optimal solution, proving the algorithm is not always correct.

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    Conditional Statement (Implication)

    A statement that asserts if one proposition (antecedent) is true, then another proposition (consequent) is also true. It's symbolized by '→'.

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    Universal Quantifier (∀)

    A logical symbol that indicates that a statement is true for every element in a given set.

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    Existential Quantifier (∃)

    A logical symbol that indicates that there exists at least one element in a given set that satisfies a statement.

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    Proof Technique

    A method used to demonstrate the truth or validity of a statement or theorem.

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    Theorem

    A statement that can be shown to be true under certain conditions.

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    Implication (if-then)

    A logical statement that asserts if one proposition (antecedent) is true, then another proposition (consequent) is also true.

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    Biconditional (if and only if)

    A logical statement that asserts two propositions are equivalent, meaning they are true or false together.

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    What makes a conditional statement FALSE?

    A conditional statement is false ONLY when the antecedent (the 'if' part) is true and the consequent (the 'then' part) is false.

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    What makes a biconditional statement FALSE?

    A biconditional statement is false if the two propositions have different truth values - one is true and the other is false.

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    Sufficient Condition

    The antecedent of a conditional statement, which, if true, guarantees the truth of the consequent.

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    Necessary Condition

    The consequent of a conditional statement, which must be true for the antecedent to be true.

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    How do you determine if a biconditional statement is true?

    To determine if a biconditional statement is true, you must check if both sides of the statement have the same truth value. If both sides are true, or both sides are false, the statement is true.

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    Direct Proof

    A proof where you logically deduce the conclusion from axioms, definitions, and previously proven theorems, directly establishing the truth of the statement.

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    Indirect Proof

    A proof that assumes the opposite of what you want to prove and then shows that this assumption leads to a contradiction, proving your original claim must be true.

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    Proof by Contradiction

    A proof technique where you assume the statement you're trying to prove is false and show that this leads to a logical contradiction, implying that your initial assumption must have been wrong, thus proving the statement.

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    What is a counterexample used for?

    A counterexample is used to demonstrate that a general statement or algorithm is not always true by providing a specific instance where the statement or algorithm fails.

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    What is a counterexample?

    A specific case or instance that contradicts or disproves a general statement or algorithm.

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    What is the use of asymptotic analysis?

    Asymptotic analysis helps you understand how an algorithm's efficiency scales with large inputs, allowing you to choose the most efficient solution for large datasets.

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    Big-Theta Notation

    A way to describe the relationship between two functions where one function is bounded, both above and below, by multiples of the other function.

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    Big-Omega Notation

    Describes a lower bound on the growth rate of a function, meaning a function grows at least as fast as a constant multiple of another function for large input values.

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    What does it mean for f(n) to be Theta of g(n)?

    It means that for large values of 'n', there exist positive constants 'c1', 'c2', and 'k' such that 'c1 * g(n)' ≤ 'f(n)' ≤ 'c2 * g(n)' whenever 'n' is greater than or equal to 'k'.

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    What are witnesses (C, k) in Big-O Notation?

    They are values (C and k) that prove a function f(x) is bounded above by a constant multiple of another function g(x), satisfying the condition |f(x)| ≤ C|g(x)| for all x>k.

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    What is the significance of Big-O Notation in computer science?

    It helps analyze and classify the efficiency of algorithms and data structures by providing a way to understand their performance as input size increases.

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    What are the key elements of a Big-Theta proof?

    It involves finding a specific value for 'k' (where the inequality applies) and finding constants 'c1' and 'c2' that satisfy the bounding condition: c1 * g(n) ≤ f(n) ≤ c2 * g(n) for all n ≥ k.

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    What is the difference between Big-O and Big-Theta notation?

    Big-O provides an upper bound on the growth rate, while Big-Theta provides both an upper and lower bound. Big-O is like saying 'no faster than', while Big-Theta is like saying 'grows at the same rate as'.

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    If and Only If (IFF)

    A biconditional statement expressing equivalence between two propositions. It's true only when both propositions have the same truth value (both true or both false).

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    Bit String

    A sequence of zeros and ones representing information.

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    Bitwise OR

    A logical operation on two bit strings, where the output bit is 1 if either or both of the corresponding input bits are 1.

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    Bitwise AND

    A logical operation on two bit strings, where the output bit is 1 only if both corresponding input bits are 1.

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    Bitwise XOR

    A logical operation on two bit strings, where the output bit is 1 only if exactly one of the corresponding input bits is 1.

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    System Specifications

    Formal descriptions of system requirements, outlining how the system should function and what features it should have.

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    Consistent Specifications

    System requirements that do not conflict with each other, ensuring a contradiction cannot be derived.

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    Applications of Propositional Logic

    Propositional logic has applications in various fields like computer science, mathematics, and software development.

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    Logic in Hardware and Software Design

    Propositional logic is used to ensure precision and consistency in defining specifications for hardware and software systems before development.

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    Witnesses (C, k) in Big-O

    Values (C and k) that prove a function f(x) is bounded above by a constant multiple of another function g(x). They show that f(x) is O(g(x)).

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    Why is Big-O notation important?

    Big-O notation helps us understand how algorithms and data structures scale with larger input sizes, allowing us to choose the most efficient solution for large datasets.

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    Example of Big-O Proof

    To prove that f(x) = x² + 2x + 1 is O(x²), we can choose k = 1 and C = 4, as this satisfies the inequality |f(x)| ≤ C|g(x)| for all x > k.

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    Study Notes

    Course Information

    • Course name: EMATH 1105
    • Course title: Discrete Mathematics I for SE
    • Prepared by: Engr. Alimo-ot & Engr. Sacramento, ECE Department

    Foundations: Logic

    • Proposition: Basic building block of logic.
    • Proposition is a declarative sentence that is either TRUE or FALSE, but not both.
    • Examples:
      • Manila is the capital of the Philippines. (TRUE)
      • Toronto is the capital of Canada. (FALSE)
      • 1 + 1 = 2. (TRUE)
      • 2 + 2 = 3. (FALSE)
    • Sentences that are not declarative are not propositions.
    • Sentences with variables are not propositions unless the variables have values assigned.

    Logical Operators

    • Operators or connectives create compound propositions from two or more propositions.
    • Negation (¬): The opposite of a proposition.
      • Examples:
        • "Michael's PC runs Linux." --> "Michael's PC does not run Linux."
        • "Julienne's smartphone has at least 32GB of memory" --> "Julienne's smartphone has less than 32GB of memory."
    • Conjunction (^): "and". True only if both propositions are true.
    • Disjunction (∨): "or". True if at least one proposition is true.
    • Exclusive or (⊕): True if exactly one proposition is true.
    • Conditional (→): "if...then". True unless a true hypothesis leads to a false conclusion.
    • Biconditional (↔): "if and only if". True only when both propositions have the same truth value.

    Logical Operators: Sample Problems

    • Identify propositions and their truth values.
    • Examples:
      • Manila is the capital of the Philippines. (TRUE)
      • Osaka is the capital of Japan. (FALSE)

    Propositional Calculus / Propositional Logic

    • The area of logic that deals with propositions.
    • Developed systematically by Aristotle over 2300 years ago.
    • Compound propositions are formed from existing propositions using logical operators.

    Truth Tables

    • Used to show the truth value of a compound proposition for all possible truth values of its components.
    • Used to show logical equivalencies.

    Inverse, Converse, Contrapositive

    • Inverse: Negates both the premise and the conclusion of a conditional statement.
    • Converse: Exchanges the premise and the conclusion of a conditional statement.
    • Contrapositive: Negates both the premise and the conclusion and exchanges them.
      • The contrapositive of a conditional statement has the same truth value as the original conditional statement.

    Logical Equivalences

    • Compound propositions with the same truth values in all cases.
    • Methods are used extensively in the construction of mathematical arguments.
    • Tautology: Always true, regardless of input.
    • Contradiction: Always false, regardless of input.
    • Contingency: Neither a tautology nor a contradiction.

    Propositional Satisfiability

    • A compound proposition is satisfiable if there is an assignment to the variables that makes the proposition true.
    • A proposition is unsatisfiable if there is no way to assign values to the variables such that the proposition is true.

    Proof Techniques

    • Direct Proof: Conclusion is established from axioms, definitions, and earlier theorems.
    • Indirect Proof: Conclusion is inferred from the premise "if not q then not p"
    • Proof by Contradiction: Shows that a logical contradiction would occur if a statement were true, thus the statement must be false.
    • Proof by Mathematical Induction: Proves a statement for infinitely many cases by proving a base case and an induction rule.
    • Proof by Construction/Example: Constructing a specific example to show a property exists or to disprove a proposition.
    • Proof by Equivalence: Showing p ↔ q by proving p → q and q → p

    Growth of a Function

    • Function's growth is determined by the highest order term.
    • Big-O Notation: Used to express the time complexity of an algorithm.
    • Big-Omega Notation: Used to define the lower bound for a function's growth.
    • Big-Theta Notation: Used to define both upper and lower bounds for a function's growth.

    Algorithm

    • A finite sequence of precise instructions for performing a computation or solving a problem.
    • Properties of algorithms: Input, output, definiteness, correctness, finiteness, effectiveness, and generality.
    • Greedy Algorithm: An algorithmic approach that makes the seemingly best choice at each step.

    Quantifiers and First-order Logic

    • Quantifiers are symbols to quantify variables.
    • Universal quantification (∀): Specifies a predicate is true for all elements in a domain.
    • Existential quantification (∃): Specifies a predicate is true for at least one element in a domain.
    • Quantifier scope: The part of the formula affected by a quantifier.
    • Free variable: A variable without a quantifier.
    • Bound variable: A variable with a quantifier.

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    Description

    This quiz will test your understanding of the foundations of logic, including the basic concepts of propositions and logical operators. Explore declarative sentences and learn how to create compound propositions using connectives. Test your knowledge and see how well you grasp these essential elements of discrete mathematics.

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