1.4 to 1.6

p ↔ ¬p Tautology Contradiction Neither a tautology or a contradiction

Contradiction

The statement ∃x P(x) asserts that

P(x) is true for at least one possible value for x in its domain.

The symbol ∃ is an

existential quantifier

Conditional identities

p → q ≡ ¬p ∨ q p ↔ q ≡ ( p → q ) ∧ ( q → p )

Commutative laws:

p ∨ q ≡ q ∨ p p ∧ q ≡ q ∧ p

he logical statement ∃x P(x) is read

"There exists an x, such that P(x)".

The logical statement ∀x P(x) is read

"for all x, P(x)" or "for every x, P(x)".

Associative laws:

( p ∨ q ) ∨ r ≡ p ∨ ( q ∨ r ) ( p ∧ q ) ∧ r ≡ p ∧ ( q ∧ r )

A compound proposition is a ____________ if the proposition is always false, regardless of the truth value of the individual propositions that occur in it.

Contradiction

Show whether each logical expression is a tautology, contradiction or neither. (p → q) ↔ (p ∧ ¬q)

Contradiction. Every truth value in the column for (p → q) ↔ (p ∧ ¬q) is F. p q (p → q) ↔ (p ∧ ¬q) T T F T F F F T F F F F

In this problem, the domain of discourse is the set of all integers. Which statements are true? If an existential statement is true, give an example. If a universal statement is false, give a counterexample. ∀x (x2 − x ≠ 0)

False. Counterexample: x = 1 or x = 0.

Prove that the following pairs of expressions are not logically equivalent. ¬p → q and ¬p ∨ q

If p = T and q = F, then ¬p → q is true and ¬p ∨ q is false. Also, when p = F and q = F, then ¬p → q is false and ¬p ∨ q is true.

(¬p ∨ q) → (p ∧ q) ¬(¬p ∨ q) ∨ (p ∧ q) (¬¬p ∧ ¬q) ∨ (p ∧ q) (p ∧ ¬q) ∨ (p ∧ q) p ∧ (¬q ∨ q) p ∧ T p

Label the steps in each proof with the law used to obtain each proposition from the previous proposition. The first line in the proof does not have a label. (¬p ∨ q) → (p ∧ q) ¬(¬p ∨ q) ∨ (p ∧ q) Conditional identity (¬¬p ∧ ¬q) ∨ (p ∧ q) De Morgan's law (p ∧ ¬q) ∨ (p ∧ q) Double negation law p ∧ (¬q ∨ q) Distributive law p ∧ T Complement law p Identity law

Determine whether the following pairs of expressions are logically equivalent. Prove your answer. If the pair is logically equivalent, then use a truth table to prove your answer. p ∧ (p → q) and p ∧ q

Logically equivalent. The columns for p ∧ (p → q) and p ∧ q are the same. p q p ∧ (p → q) p ∧ q T T T T T F F F F T F F F F F F

p → ¬p Tautology Contradiction Neither a tautology or a contradiction

Neither a tautology or a contradiction

Predicates P, T and E are defined below. The domain of discourse is the set of all positive integers. P(x): x is even T(x, y): 2x = y E(x, y, z): xy = z Indicate whether each logical expression is a proposition. If the expression is a proposition, then give its truth value. E(2, y, 7)

Not a proposition because y is a variable

Determine whether the following pairs of expressions are logically equivalent. Prove your answer. If the pair is logically equivalent, then use a truth table to prove your answer. ¬(p ∨ ¬q) and ¬p ∧ ¬q

Not logically equivalent. When p = F and q = F, then ¬(p ∨ ¬q) is false and ¬p ∧ ¬q is true. Also, when p = F and q = T, then ¬(p ∨ ¬q) is true and ¬p ∧ ¬q is false.

Predicates P, T and E are defined below. The domain of discourse is the set of all positive integers. P(x): x is even T(x, y): 2x = y E(x, y, z): xy = z Indicate whether each logical expression is a proposition. If the expression is a proposition, then give its truth value. T(5, 16) → E(6, 3, 36)

Proposition. T(5, 16) → E(6, 3, 36) is true because the hypothesis of the conditional operation, T(5, 16), is false.

Predicates P, T and E are defined below. The domain of discourse is the set of all positive integers. P(x): x is even T(x, y): 2x = y E(x, y, z): xy = z Indicate whether each logical expression is a proposition. If the expression is a proposition, then give its truth value. ¬P(3)

Proposition. ¬P(3) is true because P(3) is false.

Use the laws of propositional logic to prove that each statement is a tautology. p → (r → p)

Show that p → (r → p) ≡ T. p → (r → p) ¬p ∨ (r → p) Conditional identity ¬p ∨ (¬r ∨ p) Conditional identity ¬p ∨ (p ∨ ¬r) Commutative law (¬p ∨ p) ∨ ¬r Associative law T ∨ ¬r Complement law T Domination law

Use the laws of propositional logic to prove the following: p ∧ (¬p → q)

Solution p ∧ (¬p → q) p ∧ (¬¬p ∨ q) Conditional identity p ∧ (p ∨ q) Double negation law p Absorption law

(p ∧ q) → p Tautology Contradiction. Neither a tautology or a contradiction.

Tautology

Show whether each logical expression is a tautology, contradiction or neither. (p → q) ∨ p

Tautology. Every truth value in the column for (p → q) ∨ p is T. p q (p → q) ∨ p T T T T F T F T T F F T

In this problem, the domain of discourse is the set of all integers. Which statements are true? If an existential statement is true, give an example. If a universal statement is false, give a counterexample. ∃x (x2 > 0)

True. Example: any integer besides 0.

In this problem, the domain of discourse is the set of all integers. Which statements are true? If an existential statement is true, give an example. If a universal statement is false, give a counterexample. ∃x (x + 2 = 1)

True. Example: x = -1.

Prove that the following pairs of expressions are not logically equivalent. p ∧ (p → q) and p ∨ q

When p = T and q = F (or when p = F and q = T), then p ∧ (p → q) is false and p ∨ q is true.

A counterexample for a universally quantified statement is

an element in the domain for which the predicate is false. A single counterexample is sufficient to show that a universally quantified statement is false.

the statement ∃x P(x) is called a

existentially quantified statement

logically equivalent

if they have the same truth value regardless of the truth values of their individual propositions.

The domain of a variable in a predicate

is the set of all possible values for the variable. For example, a natural domain for the variable x in the predicate "x is an odd number" would be the set of all integers.

Use the laws of propositional logic to prove the following: p ↔ (p ∧ r) ≡ ¬p ∨ r

p ↔ (p ∧ r) (p → (p ∧ r)) ∧ ((p ∧ r) → p) Conditional identity (p → (p ∧ r)) ∧ (¬(p ∧ r) ∨ p) Conditional identity (p → (p ∧ r)) ∧ ((¬p ∨ ¬r) ∨ p) De Morgan's law (p → (p ∧ r)) ∧ ((¬r ∨ ¬p) ∨ p) Commutative law (p → (p ∧ r)) ∧ (¬r ∨ (¬p ∨ p)) Associative law (p → (p ∧ r)) ∧ (¬r ∨ T) Complement law (p → (p ∧ r)) ∧ T Domination law p → (p ∧ r) Identity law ¬p ∨ (p ∧ r) Conditional identity (¬p ∨ p) ∧ (¬p ∨ r) Distributive law T ∧ (¬p ∨ r) Complement law ¬p ∨ r Identity law

Complement laws:

p ∧ ¬p ≡ F ¬T ≡ F p ∨ ¬p ≡ T ¬F ≡ T

Distributive laws:

p ∨ ( q ∧ r ) ≡ ( p ∨ q ) ∧ ( p ∨ r ) p ∧ ( q ∨ r ) ≡ ( p ∧ q ) ∨ ( p ∧ r )

Translate each English sentence into a logical expression using the propositional variables defined below. Then negate the entire logical expression using parentheses and the negation operation. Apply De Morgan's law to the resulting expression and translate the final logical expression back into English. p: the applicant has written permission from his parents e: the applicant is at least 18 years old s: the applicant is at least 16 years old The applicant has written permission from his parents or is at least 18 years old.

p ∨ e ¬(p ∨ e) ¬p ∧ ¬e The applicant does not have written permission from his parents and is not at least 18 years old.

A logical statement whose truth value is a function of one or more variables is called a

predicate

A compound proposition is a _________ if the proposition is always true, regardless of the truth value of the individual propositions that occur in it.

tautology

De Morgan's laws:

¬( p ∨ q ) ≡ ¬p ∧ ¬q ¬( p ∧ q ) ≡ ¬p ∨ ¬q

Use the laws of propositional logic to prove the following: ¬(p ∨ (¬p ∧ q)) ≡ ¬p ∧ ¬q

¬(p ∨ (¬p ∧ q)) (¬p ∧ ¬(¬p ∧ q)) De Morgan's law (¬p ∧ (¬¬p ∨ ¬q)) De Morgan's law (¬p ∧ (p ∨ ¬q)) Double negation law (¬p ∧ p) ∨ (¬p ∧ ¬q) Distributive law F ∨ (¬p ∧ ¬q) Complement law ¬p ∧ ¬q Identity law

The first De Morgan's law is:

¬(p ∨ q) ≡ (¬p ∧ ¬q)

Use the laws of propositional logic to prove the following: ¬p → (q → r) ≡ q → (p ∨ r)

¬p → (q → r) ¬¬p ∨ (q → r) Conditional identity p ∨ (q → r) Double negation law p ∨ (¬q ∨ r) Conditional identity (p ∨ ¬q) ∨ r Associative law (¬q ∨p) ∨ r Commutative law ¬q ∨ (p ∨ r) Associative law q → (p ∨ r) Conditional identity

Absorption laws:

p ∨ (p ∧ q) ≡ p p ∧ (p ∨ q) ≡ p

Identity laws:

p ∨ F ≡ p p ∧ T ≡ p

Idempotent laws:

p ∨ p ≡ p p ∧ p ≡ p

Show whether each logical expression is a tautology, contradiction or neither. (¬p ∨ q) ↔ (¬p ∧ q)

Neither. If p = F and q = T, then (¬p ∨ q) ↔ (¬p ∧ q) is true. If p = T and q = T, then (¬p ∨ q) ↔ (¬p ∧ q) is false.

Predicates P, T and E are defined below. The domain of discourse is the set of all positive integers. P(x): x is even T(x, y): 2x = y E(x, y, z): xy = z Indicate whether each logical expression is a proposition. If the expression is a proposition, then give its truth value. T(5, x)

Not a proposition because x is a variable

Select the English sentence that is logically equivalent to the given sentence. It is not true that the child is at least 8 years old or at least 57 inches tall. The child is at least 8 years old or at least 57 inches tall. The child is less than 8 years old or shorter than 57 inches. The child is less than 8 years old and shorter than 57 inches.

The child is less than 8 years old and shorter than 57 inches.

Select the English sentence that is logically equivalent to the given sentence. 1) It is not true that the child is at least 8 years old and at least 57 inches tall. The child is at least 8 years old and at least 57 inches tall. The child is less than 8 years old or shorter than 57 inches. The child is less than 8 years old and shorter than 57 inches.

The child is less than 8 years old or shorter than 57 inches.

De Morgan's laws

are logical equivalences that show how to correctly distribute a negation operation inside a parenthesized expression.

Use truth tables to show that the following pairs of expressions are logically equivalent. ¬(p ↔ q) and ¬p ↔ q

p q ¬(p ↔ q) ¬p ↔ q T T F F T F T T F T T T F F F F

Domination laws:

p ∧ F ≡ F p ∨ T ≡ T

The second version of De Morgan's law swaps the role of the disjunction and conjunction:

¬(p ∧ q) ≡ (¬p ∨ ¬q)

Double negation law:

¬¬p ≡ p

universal quantifier

∀

Consider the following statements in English. Write a logical expression with the same meaning. The domain of discourse is the set of all real numbers. Every number is less than or equal to its square.

∀x (x ≤ x2)

Consider the following statements in English. Write a logical expression with the same meaning. The domain of discourse is the set of all real numbers. The square of every number is at least 0.

∀x (x2 ≥ 0)

universally quantified statement

∀x P(x)