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

What is a Boolean variable?

A variable that can be either true or false.

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

0, 1 Signup and view all the answers

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

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

Match the logical operator with its description:

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

What is the symbol used to denote 'negation'?

¬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'.

True Signup and view all the answers

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

True Signup and view all the answers

What is represented by 1 and 0 in Boolean logic?

1 represents true, and 0 represents false. Signup and view all the answers

What does the symbol 'XOR' stand for?

exclusive or Signup and view all the answers

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

p ∧ q Signup and view all the answers

Match the following logical operations with their definitions:

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 Signup and view all the answers

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

There is pollution in Iloilo. Signup and view all the answers

The biconditional operation is denoted by p ↔ q.

True Signup and view all the answers

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

true Signup and view all the answers

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

True Signup and view all the answers

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

False Signup and view all the answers

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

False Signup and view all the answers

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

False Signup and view all the answers

Match the following expressions with their expected outcomes:

$x ∧ y$ = 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 Signup and view all the answers

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

False Signup and view all the answers

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

False Signup and view all the answers

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

False Signup and view all the answers

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

False Signup and view all the answers

What is the origin of the term 'algorithm'?

A mathematician's name Signup and view all the answers

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

False Signup and view all the answers

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

Correctness Signup and view all the answers

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

greedy Signup and view all the answers

Match the properties of algorithms with their definitions:

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

Which of the following is NOT a property of algorithms?

Infinity Signup and view all the answers

The steps of an algorithm must be defined precisely.

True Signup and view all the answers

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

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

What does a compound proposition need to be considered satisfiable?

There must be at least one assignment of truth values that makes it true. Signup and view all the answers

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

True Signup and view all the answers

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

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

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

false Signup and view all the answers

Match the following terms with their definitions:

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 Signup and view all the answers

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

Quantifiers help define the range of variables in logical expressions. Signup and view all the answers

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

True Signup and view all the answers

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

p Signup and view all the answers

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

To demonstrate that f(x) grows slower than g(x) 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).

False Signup and view all the answers

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

Constants C and k Signup and view all the answers

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

1892 Signup and view all the answers

Match the following big-O concepts with their definitions:

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 Signup and view all the answers

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

1 Signup and view all the answers

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

True Signup and view all the answers

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

4x^2 Signup and view all the answers

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

False Signup and view all the answers

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

True Signup and view all the answers

What is the definition of the inverse in logic?

The inverse is formed by negating both the hypothesis and the conclusion of a conditional statement. 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 _____'.

it rains Signup and view all the answers

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

¬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.

True Signup and view all the answers

Provide an example of a statement and its contrapositive.

'If A, then B' and 'If not B, then not A'. Signup and view all the answers

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

If p then q. Signup and view all the answers

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

True Signup and view all the answers

What is the definition of a biconditional statement?

A biconditional statement is 'p if and only if q'. 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.

true Signup and view all the answers

Match each implication with its equivalent phrase:

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

Which of the following is NOT true regarding implications?

If both p and q are false, the implication is false. 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.

False Signup and view all the answers

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

q Signup and view all the answers

What is a bound variable in the context of quantifiers?

A variable that is associated with a quantifier. 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.

True Signup and view all the answers

What is the scope of a quantifier?

The formula immediately following the quantifier. Signup and view all the answers

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

bound Signup and view all the answers

Match the expression to its quantifier:

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

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

P(x) is true for all values of x. 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.

True Signup and view all the answers

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

∃x P(x) Q(x) Signup and view all the answers

What characterizes a greedy algorithm?

It picks the best option available at each step Signup and view all the answers

A greedy algorithm is guaranteed to find an optimal solution.

False Signup and view all the answers

What is the purpose of asymptotic analysis?

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

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

asymptotic Signup and view all the answers

Match the following functions with their characteristics:

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 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?

Program A 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 $g(x)$.

False Signup and view all the answers

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

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

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

If today isn't Friday, then I don't have a test today. 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.'

True Signup and view all the answers

What logical operator is used to denote 'and'?

∧ Signup and view all the answers

In Boolean logic, the value _______ represents true.

1 Signup and view all the answers

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

False Signup and view all the answers

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

True Signup and view all the answers

What is the purpose of constructing a truth table?

To determine the truth values of compound propositions. Signup and view all the answers

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

False Signup and view all the answers

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

False Signup and view all the answers

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

~𝑎 Signup and view all the answers

The disjunction of 𝑎 and 𝑏 is denoted as ______.

𝑎 ∨ 𝑏 Signup and view all the answers

Match the following logical operations with their truth table outcomes:

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 Signup and view all the answers

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

True if a is true, regardless of b Signup and view all the answers

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

False 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 ______.

false Signup and view all the answers

When is a conditional statement p → q considered false?

When p is true and q is false 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.

True Signup and view all the answers

What does the XOR operator signify in logical operations?

It signifies 'exclusive or', meaning one condition is true, but not both. 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 $q$.

true Signup and view all the answers

Which of the following statements represents a conditional statement?

If it rains, then the ground will be wet. 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.

False Signup and view all the answers

The symbol that represents a conditional statement is _____ .

→ Signup and view all the answers

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

Direct Proof Signup and view all the answers

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

True Signup and view all the answers

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

If x is odd, then x^2 is odd. 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.

contradiction Signup and view all the answers

Match the following proof techniques with their descriptions:

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 Signup and view all the answers

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

p only if q 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?

If x is not even, then x² is not even. 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'?

If I have a test today, then today is Friday. 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.

False Signup and view all the answers

The product of two odd numbers is __________.

odd Signup and view all the answers

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

false 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.'

False Signup and view all the answers

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

Prove that 2 is irrational. Signup and view all the answers

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

If I don't have a test today, then today is not Friday. 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 ↔ q Signup and view all the answers

Match the following logical implications with their meanings:

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 Signup and view all the answers

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

true, false Signup and view all the answers

Which of the following statements is true regarding conditional statements?

A false premise allows for any conclusion to be valid. Signup and view all the answers

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

True Signup and view all the answers

What does the symbol ~ represent in Boolean logic?

Negation Signup and view all the answers

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

↔ Signup and view all the answers

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

false Signup and view all the answers

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

True Signup and view all the answers

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

For all Signup and view all the answers

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

False Signup and view all the answers

What is a theorem?

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

The squares of all ______ integers are odd.

odd Signup and view all the answers

Match the following logical statements with their types:

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 Signup and view all the answers

Which statement correctly reflects the meaning of the symbol ∃?

There exists at least one Signup and view all the answers

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

True Signup and view all the answers

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

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

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

f(x) is bounded both above and below by g(x) Signup and view all the answers

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

True Signup and view all the answers

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

3 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 _____.

n ≥ k Signup and view all the answers

Match the following functions with their corresponding asymptotic notations:

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

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

6n Signup and view all the answers

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

False Signup and view all the answers

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

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

What is a key characteristic of a greedy algorithm?

It finds a feasible solution quickly. Signup and view all the answers

Greedy algorithms are also known as the shortest path algorithms.

True Signup and view all the answers

What is the main concept of asymptotic analysis?

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

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

30n + 8 Signup and view all the answers

Which of the following statements about logical implications is true?

Only the contrapositive has the same truth value as the original implication. Signup and view all the answers

Match the following growth rates with their descriptions:

fA(n) = Linear growth fB(n) = Quadratic growth Signup and view all the answers

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

f(x) will eventually become larger than g(x) as x grows. Signup and view all the answers

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

False Signup and view all the answers

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

true Signup and view all the answers

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

False Signup and view all the answers

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

It is snowing and it is not cold Signup and view all the answers

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

Program A Signup and view all the answers

Match the terms with their definitions:

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 Signup and view all the answers

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

False Signup and view all the answers

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

If today isn't Friday, then I don't have a test today. Signup and view all the answers

What does the statement $p ⊕ q$ represent?

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

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

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

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

False Signup and view all the answers

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

today Signup and view all the answers

Match the logical operators with their definitions:

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

Which of the following correctly describes a Boolean variable?

A variable that can be true or false. Signup and view all the answers

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

False Signup and view all the answers

The symbol for logical implication is ______.

→ Signup and view all the answers

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

f(x) must be O(g(x)) and Ω(g(x)) Signup and view all the answers

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

True 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?

C1 = 4, C2 = 6 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 = _.

lower bound, upper bound Signup and view all the answers

Match the following expressions with their meanings:

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 Signup and view all the answers

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

There exists an assignment of truth values that makes it true. Signup and view all the answers

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

False Signup and view all the answers

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

To specify a point from which the inequality holds Signup and view all the answers

What operation does the symbol ⊕ represent in propositional logic?

Bitwise XOR Signup and view all the answers

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

p Signup and view all the answers

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

False Signup and view all the answers

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

True Signup and view all the answers

What is the relationship indicated by Big-Theta notation?

Tight bound between two functions Signup and view all the answers

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

tautology Signup and view all the answers

Match the logical terms with their definitions:

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 Signup and view all the answers

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

4 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.

OR Signup and view all the answers

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

Distributive Signup and view all the answers

The expression ¬q ∧ q is always true.

False Signup and view all the answers

Match the following bitwise operations with their outcomes:

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 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 __________.

truth values Signup and view all the answers

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

0110 Signup and view all the answers

Inconsistent system specifications can lead to successful software development.

False Signup and view all the answers

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

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

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.

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'?

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

What is a Boolean variable?

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

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

Match the logical operator with its description:

What is the symbol used to denote 'negation'?

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

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

What is represented by 1 and 0 in Boolean logic?

What does the symbol 'XOR' stand for?

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

Match the following logical operations with their definitions:

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

The biconditional operation is denoted by p ↔ q.

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

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

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

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

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

Match the following expressions with their expected outcomes:

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

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

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

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

What is the origin of the term 'algorithm'?

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

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

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

Match the properties of algorithms with their definitions:

Which of the following is NOT a property of algorithms?

The steps of an algorithm must be defined precisely.

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

What does a compound proposition need to be considered satisfiable?

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

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

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

Match the following terms with their definitions:

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

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

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

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

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

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

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

Match the following big-O concepts with their definitions:

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

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

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

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

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

What is the definition of the inverse in logic?

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

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

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

Provide an example of a statement and its contrapositive.

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

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

What is the definition of a biconditional statement?

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

Match each implication with its equivalent phrase:

Which of the following is NOT true regarding implications?

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

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

What is a bound variable in the context of quantifiers?

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

What is the scope of a quantifier?

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

Match the expression to its quantifier:

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

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

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

What characterizes a greedy algorithm?

A greedy algorithm is guaranteed to find an optimal solution.

What is the purpose of asymptotic analysis?

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

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

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