EMath 1105 Discrete Mathematics I: Function Growth

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Questions and Answers

What is the negation of the statement 'All integers are odd and prime'?

Some integers are odd and prime.

No integers are odd and prime.

All integers are not primes.

Not all integers are odd and prime. (correct)

Which of the following statements aligns with the expression ∀x (P(x) → R(x))?

If x is an odd integer, then x² is odd for all integers. (correct)

All odd integers have even squares.

For some integers, if x is odd, then x² is odd.

There exists an integer x such that if x is odd, then x² is odd.

What is the English statement that corresponds to ∃x (P(x) ∧ Q(x))?

All prime integers are odd.

Every odd integer is also a prime integer.

Some odd integers are prime. (correct)

No odd integers are prime.

Which of the following represents a conjunction?

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

How would you represent the statement 'If x is odd, then x² is odd' using logical symbols?

P(x) → R(x) Signup and view all the answers

Which of the following statements is true about the truth table of P(x) ∧ Q(x)?

The statement is true when both P and Q are true. Signup and view all the answers

What is the correct description of the quantifier ∃ in logical statements?

It indicates the existence of at least one element satisfying a condition. Signup and view all the answers

Which logical operation is expressed with the symbol ∨?

Disjunction Signup and view all the answers

What is required to prove a statement of the form p q using proof by equivalence?

Both p → q and q → p must be true Signup and view all the answers

Which of the following statements uses negation correctly?

Not all integers are rational numbers. Signup and view all the answers

In the example proving that the product of even integers is even, which logical operation is primarily used?

Implication Signup and view all the answers

What conclusion can be drawn from the statement "Every positive integer is the sum of the squares of two integers" using proof by counterexample?

It is false for at least one integer. Signup and view all the answers

What is a truth table primarily used for in logic?

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

Which of the following best represents a proposition?

All apples are green. Signup and view all the answers

What is the primary characteristic of propositional logic?

It evaluates the truth and falsity of statements only. Signup and view all the answers

What is the correct negation of the proposition Q(2,3)?

Q(2,3) is false. Signup and view all the answers

Given the proposition P: x² is greater than or equal to y, which of the following is an example of conjunction involving this proposition?

P and Q(3,2) where Q(3,2) is true. Signup and view all the answers

What defines the correctness property of an algorithm?

An algorithm should produce the correct output values for each set of input values. Signup and view all the answers

Which property of algorithms ensures that the steps are clearly definable?

Definiteness Signup and view all the answers

In truth tables, when will a conjunction P ∧ Q evaluate to true?

When both P and Q are true. Signup and view all the answers

What is the primary goal of an algorithm in terms of input and output?

To produce output values that solve the problem based on the given input. Signup and view all the answers

Which of the following correctly represents a false statement based on the example of Q(x,y)?

Q(5,25) is false. Signup and view all the answers

How many distinct output values can an algorithm yield from a specific set of input values?

At least one output value must be generated. Signup and view all the answers

What is a theorem in the context of mathematics and computer science?

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

Which logical statement correctly expresses that all squares of odd integers are odd?

∀x (P(x) → R(x)) Signup and view all the answers

What does the quantifier ∀ represent in logical expressions?

All elements satisfy a condition. Signup and view all the answers

Which of the following statements is correct regarding the expression ∃x (P(x) ∧ Q(x))?

Some odd integers are prime. Signup and view all the answers

Which logical operation's results align with the statement 'If x is odd, then x² is odd'?

Implication Signup and view all the answers

Which of the following describes a typical step in proving a theorem?

Identifying conditions under which the theorem holds. Signup and view all the answers

What is the primary goal of an algorithm?

To produce a result from given input. Signup and view all the answers

In proof techniques, which form of reasoning often strengthens an argument?

Generalization of specific cases. Signup and view all the answers

Which statement accurately describes proof by counterexample?

It is used to provide a specific instance that disproves a statement. Signup and view all the answers

What is the main conclusion from the product of two even integers?

The product is an even integer. Signup and view all the answers

In first-order logic, what must be true for the statement 'Every integer is a rational number' to be valid?

All integers can be represented as fractions. Signup and view all the answers

Which of the following describes the structure of a statement in propositional logic?

It focuses only on the truth or falsity of the statement. Signup and view all the answers

Which operation is necessary to prove a statement of the form p → q using proof by equivalence?

Proving both p → q and q → p. Signup and view all the answers

What is a key aspect of quantifiers in first-order logic?

They allow for statements about all or some elements of a set. Signup and view all the answers

What type of integer is necessary when proving the product of two integers is even?

Any integers, positive or negative. Signup and view all the answers

In proof techniques, which is the result of using proof by construction?

Providing an example that demonstrates a concept. Signup and view all the answers

Which of the following correctly describes the property of finiteness in algorithms?

An algorithm must complete after a finite number of steps. Signup and view all the answers

What does the property of definiteness in an algorithm entail?

The algorithm must have a clear and precise sequence of steps. Signup and view all the answers

In the context of first-order logic, what does the symbol ∀ represent?

All elements in the domain are included. Signup and view all the answers

If Q(x, y) states that $x^2$ is greater than or equal to y, which of the following pairs would falsify the statement?

(3, 8) Signup and view all the answers

What distinguishes correctness as a property of algorithms?

An algorithm must produce accurate output values for each set of inputs. Signup and view all the answers

What does the term 'quantifier' in logic refer to?

A symbol that specifies the quantity of subjects in a statement. Signup and view all the answers

Which of the following defines an algorithm's input?

The values and conditions that the algorithm processes. Signup and view all the answers

In first-order logic, if Q(2, 5) is false, which of the following conclusions can be drawn?

2 squared is less than or greater than 5. Signup and view all the answers

What does the statement ∀x (P(x) → R(x)) imply about integers?

All odd integers have odd squares. Signup and view all the answers

Which description aligns with the theorem that if x is odd, then x² is odd?

It is true for all integers. Signup and view all the answers

Which type of quantifier is represented by the statement ∃x (P(x) ∧ Q(x))?

Existential quantifier Signup and view all the answers

What is essential for proving the correctness of an algorithm?

The steps must be defined and repeatable. Signup and view all the answers

In logical statements, what does the symbol ∧ represent?

Logical conjunction Signup and view all the answers

How can a proof by counterexample affect a logical statement?

It provides evidence that the statement is false. Signup and view all the answers

What can be concluded from the statement 'All squares of odd integers are odd'?

All odd integers have odd squares. Signup and view all the answers

What role do logical connectives play in first-order logic?

They combine statements into more complex logical expressions. Signup and view all the answers

What is the key purpose of using proof by counterexample?

To disprove a statement by providing a specific instance where it fails Signup and view all the answers

Which of the following definitions correctly describes a bi-conditional statement?

A statement that states p if and only if q is true Signup and view all the answers

In the context of quantifiers in first-order logic, which statement is accurate regarding universal quantification?

It allows for the assertion that a property holds for all members of a domain Signup and view all the answers

When proving that the product of two integers is even, which logical reasoning is primarily utilized?

Direct proof demonstrating a relationship between even integers Signup and view all the answers

What characterizes the concept of first-order logic compared to propositional logic?

First-order logic incorporates quantifiers and relationships between objects Signup and view all the answers

What defines the 'Tautology' in the context of proof methods?

A statement that is false under no circumstance Signup and view all the answers

In algorithm analysis, which property is essential for ensuring an algorithm yields correct results?

Finiteness in execution steps Signup and view all the answers

Which of the following illustrates an incorrect use of quantifiers in logical statements?

For every positive integer, there exists a prime number Signup and view all the answers

What must hold true for the correctness property of an algorithm?

The algorithm should yield the correct output for every possible set of input values. Signup and view all the answers

Which property of algorithms states that they must terminate after a finite number of steps?

Finiteness Signup and view all the answers

In the example Q(2, 3) is true, which of the following statements is correct regarding Q(2, 5)?

Q(2, 5) is false because 4 is neither greater nor equal to 5. Signup and view all the answers

What is an essential requirement for the input property of algorithms?

The input values should come from a specified set. Signup and view all the answers

What does definiteness mean in relation to an algorithm's steps?

The steps should be precisely defined to avoid ambiguity. Signup and view all the answers

How can you express that a property P is true for all integers using a quantifier?

∀x P(x) Signup and view all the answers

What is the main concept of an algorithm defined as?

A finite sequence of definitive instructions. Signup and view all the answers

If Q(x, y) is defined as 'x² is greater than or equal to y', which statement logically follows from Q(3, 9)?

Q(3, 9) is true since 9 is equal to 9. Signup and view all the answers

Study Notes

EMath 1105 Discrete Mathematics I for SE

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

Growth of a Function

Function growth determined by highest order term
For large input values, function grows as fast as the largest term
Constant multiples don't significantly affect the growth rate
Primarily concerned with the shape of the curve, not the slope or y-intercept
Asymptotic analysis computes running time of operations in mathematical units

Big-O Notation

Used to express time complexity of an algorithm
Compares efficiency of algorithms as input size grows
Assumes each operation takes the same amount of time
Time complexity independent of software/hardware used

Big-O Notation (Example)

If algorithm 1 uses 100n² + 17n + 4 operations and algorithm 2 uses n³ operations
Algorithm 1 will be more efficient for large input sizes.

Big-O Notation (Formal Definition)

f(x) is O(g(x)) if there exist constants C and k such that |f(x)| ≤ C|g(x)| for all x > k

Big-O Notation (Finding Witnesses)

To prove f(x) is O(g(x)), select a value of k
Determine how to estimate |f(x)| for x > k
Determine a value of C

Big-O Notation (Example)

Show f(x) = x² + 2x + 1 is O(x²)
Values for C and k are 4 & 1 respectively

Big-Omega Notation

f(x) is Ω(g(x)) if there exist positive constants C and k such C|g(x)| ≤ |f(x)| for all x > k;
C |g(x)| is a lower bound for |f(x)|

Big-Theta Notation

f(x) is θ(g(x)) if f(x) is O(g(x)) and f(x) is Ω(g(x))
C₁|g(x)| ≤ |f(x)| ≤ C₂|g(x)| for all x>k

Big-Theta Example (Example)

Let f(n) = 4n + 6
f(n) = θ(g(n)). Find C₁, C₂, and k.
4n ≤ 4n + 6 ≤ 6n for n ≥ 3, so C₁ = 4, C₂=6, with k = 3.

Big Summary

Upper Bound - use Big-O
Lower Bound - use Big-Omega
Order of Growth - use Big-Theta

Quantifiers and First-Order Logic

Quantifiers quantify variables in predicates
Universal quantification: ∀x P(x): P(x) holds for all x within a domain
Existential quantification: ∃x P(x) : ∃x such that P(x) is true within a domain

Proof Techniques

Algorithms proven logically based on axioms, definitions, and earlier theorems
Theorems are statements proven true under given conditions
Direct proof: uses axioms and definitions

Proof by Contradiction

Demonstrates a statement is false by showing a logical contradiction

Proof by Mathematical Induction

A method to prove cases by first proving a base case then an induction step
The induction rule establishes the next cases repeating from the base case

Proof by Construction/Proof by Example

Provides a concrete example to show something having a particular property exists
May also be used to find a counter example to disprove a proposition

Proof by Equivalence/Bi-conditional Proof

Show that the statements are equivalent by showing p → q and q → p

Propositional Logic

Area of logic studying propositions' properties such as: truth and falsity, and compound propositions
Compound propositions formed from propositions using logical operators
Negation, "and", "or", exclusive or, conditional/implication, and biconditional

Logical Equivalences

Statements have same truth values in all cases
Tautology: always true
Contradiction: always false
Contingency: neither tautology nor contradiction

Propositional Satisfiability

Compound propositions are satisfiable if there exist truth values assigning to variables which makes the proposition true. Those values make it a solution to a satisfiability problem
Compound propositions are unsatisfiable if it's false for every assignment of truth values

Algorithm Properties

Input: values from a specific set for the algorithm to process
Output: values from a specified set resulting from input processes; these are the solutions to the problem.
Definiteness: steps must be defined precisely
Correctness: algorithm must produce the correct output for input sets
Finiteness: algorithm must produce output after a finite set of steps regardless of large input sets.
Effectiveness: each step must be possible to perform in a finite/limited time
Generality: procedure must be applicable to all problems in a form and not just a subset or a specific set of inputs

Greedy Algorithm

Simplest approach to optimization problems
Selects best choice at each step instead of all possible sequences which may lead to optimal solutions
Algorithms making the "best" choice at each step are called greedy algorithms

References

Rosen, K. H. (2012). Discrete Mathematics and Its Applications, 8th Edition: McGrawHill

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This quiz covers key concepts from EMath 1105 Discrete Mathematics I, focusing on the growth of functions and Big-O notation. It emphasizes function growth determined by the highest order term and explores computational analysis of algorithm efficiency. Prepare to test your understanding of time complexity and algorithm comparison.

EMath 1105 Discrete Mathematics I: Function Growth

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Questions and Answers

What is the negation of the statement 'All integers are odd and prime'?

Which of the following statements aligns with the expression ∀x (P(x) → R(x))?

What is the English statement that corresponds to ∃x (P(x) ∧ Q(x))?

Which of the following represents a conjunction?

How would you represent the statement 'If x is odd, then x² is odd' using logical symbols?

Which of the following statements is true about the truth table of P(x) ∧ Q(x)?

What is the correct description of the quantifier ∃ in logical statements?

Which logical operation is expressed with the symbol ∨?

What is required to prove a statement of the form p q using proof by equivalence?

Which of the following statements uses negation correctly?

In the example proving that the product of even integers is even, which logical operation is primarily used?

What conclusion can be drawn from the statement "Every positive integer is the sum of the squares of two integers" using proof by counterexample?

What is a truth table primarily used for in logic?

Which of the following best represents a proposition?

What is the primary characteristic of propositional logic?

What is the correct negation of the proposition Q(2,3)?

Given the proposition P: x² is greater than or equal to y, which of the following is an example of conjunction involving this proposition?

What defines the correctness property of an algorithm?

Which property of algorithms ensures that the steps are clearly definable?

In truth tables, when will a conjunction P ∧ Q evaluate to true?

What is the primary goal of an algorithm in terms of input and output?

Which of the following correctly represents a false statement based on the example of Q(x,y)?

How many distinct output values can an algorithm yield from a specific set of input values?

What is a theorem in the context of mathematics and computer science?

Which logical statement correctly expresses that all squares of odd integers are odd?

What does the quantifier ∀ represent in logical expressions?

Which of the following statements is correct regarding the expression ∃x (P(x) ∧ Q(x))?

Which logical operation's results align with the statement 'If x is odd, then x² is odd'?

Which of the following describes a typical step in proving a theorem?

What is the primary goal of an algorithm?

In proof techniques, which form of reasoning often strengthens an argument?

Which statement accurately describes proof by counterexample?

What is the main conclusion from the product of two even integers?

In first-order logic, what must be true for the statement 'Every integer is a rational number' to be valid?

Which of the following describes the structure of a statement in propositional logic?

Which operation is necessary to prove a statement of the form p → q using proof by equivalence?

What is a key aspect of quantifiers in first-order logic?

What type of integer is necessary when proving the product of two integers is even?

In proof techniques, which is the result of using proof by construction?

Which of the following correctly describes the property of finiteness in algorithms?

What does the property of definiteness in an algorithm entail?

In the context of first-order logic, what does the symbol ∀ represent?

If Q(x, y) states that $x^2$ is greater than or equal to y, which of the following pairs would falsify the statement?

What distinguishes correctness as a property of algorithms?

What does the term 'quantifier' in logic refer to?

Which of the following defines an algorithm's input?

In first-order logic, if Q(2, 5) is false, which of the following conclusions can be drawn?

What does the statement ∀x (P(x) → R(x)) imply about integers?

Which description aligns with the theorem that if x is odd, then x² is odd?

Which type of quantifier is represented by the statement ∃x (P(x) ∧ Q(x))?

What is essential for proving the correctness of an algorithm?

In logical statements, what does the symbol ∧ represent?

How can a proof by counterexample affect a logical statement?

What can be concluded from the statement 'All squares of odd integers are odd'?

What role do logical connectives play in first-order logic?

What is the key purpose of using proof by counterexample?

Which of the following definitions correctly describes a bi-conditional statement?

In the context of quantifiers in first-order logic, which statement is accurate regarding universal quantification?

When proving that the product of two integers is even, which logical reasoning is primarily utilized?

What characterizes the concept of first-order logic compared to propositional logic?

What defines the 'Tautology' in the context of proof methods?

In algorithm analysis, which property is essential for ensuring an algorithm yields correct results?

Which of the following illustrates an incorrect use of quantifiers in logical statements?

What must hold true for the correctness property of an algorithm?

Which property of algorithms states that they must terminate after a finite number of steps?

In the example Q(2, 3) is true, which of the following statements is correct regarding Q(2, 5)?

What is an essential requirement for the input property of algorithms?

What does definiteness mean in relation to an algorithm's steps?

How can you express that a property P is true for all integers using a quantifier?

What is the main concept of an algorithm defined as?

If Q(x, y) is defined as 'x² is greater than or equal to y', which statement logically follows from Q(3, 9)?

Study Notes

EMath 1105 Discrete Mathematics I for SE

Growth of a Function

Big-O Notation

Big-O Notation (Example)