04 Logical Equivalence + Conditional Statements

FALSE

A contradiction is a statement that is always ___

TRUE

A tautology is a statement that is always ____

Absorption laws: p ∨ (p ∧ q) ≡ p p ∧ (p ∨ q) ≡ p

Absorption laws: p ∨ (p ∧ q) ≡ __ p ∧ (p ∨ q) ≡ __

An or statement is false if, and only if, both components are FALSE

An OR statement is false if, and only if, both components are ____

An if-then statement is false if, and only if, the hypothesis is TRUE and the conclusion is FALSE

An if-then (p → q) statement is false if, and only if, the hypothesis is ____ and the conclusion is ____

Associative laws: (p ∧ q) ∧ r ≡ p ∧ (q ∧ r) (p ∨ q) ∨ r ≡ p ∨ (q ∨ r)

Associative laws: (p ∧ q) ∧ r ≡ ______ (p ∨ q) ∨ r ≡ ______

Commutative laws: p ∧ q ≡ q ∧ p p ∨ q ≡ q ∨ p

Commutative laws: p ∧ q ≡ ____ p ∨ q ≡ _____

∼(p ∧ q) ≡ ∼p ∨ ∼q The negation of an and statement is logically equivalent to the or statement in which each component is negated. ∼(p ∨ q) ≡ ∼p ∧ ∼q. The negation of an or statement is logically equivalent to the and statement in which each component is negated.

De Morgan's Law: ∼(p ∧ q) ≡ ... ∼(p ∨ q) ≡ ....

Distributive laws: p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r) p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)

Distributive laws: p ∧ (q ∨ r) ≡ _______ p ∨ (q ∧ r) ≡ _______

Double negative law: ∼(∼p) ≡ p

Double negative law: ∼(∼p) ≡ __

Given statement variables p and q, the biconditional of p and q is "p if, and only if, q" and is denoted p ↔ q. It is true if both p and q have the same truth values and is false if p and q have opposite truth values

Given statement variables p and q, the BICONDITIONAL of p and q is "p if, and only if, q" and is denoted p ↔ q. It is ____ if both p and q have the same truth values and is _____ if p and q have opposite truth values

Idempotent laws: p ∧ p ≡ p p ∨ p ≡ p

Idempotent laws: p ∧ p ≡ __ p ∨ p ≡ __

Identity laws: p ∧ t ≡ p p ∨ c ≡ p

Identity laws: p ∧ t ≡ ____ p ∨ c ≡ ____

Negation laws: p ∨ ∼p ≡ t p ∧ ∼p ≡ c

Negation laws: p ∨ ∼p ≡ __ p ∧ ∼p ≡ __

∼(p → q) ≡ p ∧ ∼q

Negation of a conditional statement: ∼(p → q) ≡ ____

Negations of t and c: ∼t ≡ c ∼c ≡ t

Negations of t and c: ∼t ≡ __ ∼c ≡ __

The contrapositive of a conditional statement of the form "If p then q" is If ∼q then ∼p. Symbolically, The contrapositive of p → q is ∼q → ∼p. A conditional statement is logically equivalent to its contrapositive.

The contrapositive of a conditional statement of the form "If p then q" is __________

The converse of p → q is q → p converse is not logically equivalent to the statement.

The converse of p → q is ____

The inverse of p → q is ∼p → ∼q inverse is not logically equivalent to the statement.

The inverse of p → q is ______

Two statement forms are logically equivalent if, and only if, they always have the same truth values denoted P ≡ Q

Two statement forms are logically equivalent if, and only if, they always have ____ denoted P ≡ Q

Universal bound laws: p ∨ t ≡ t p ∧ c ≡ c

Universal bound laws: p ∨ t ≡ __ p ∧ c ≡ __

p → q ≡ ∼q → ∼p ≡ q ∨ ∼p ≡ ∼p ∨ q

p → q ≡ ____ ≡ _____ ≡ _____

p ↔ q ≡ (p → q) ∧ (q → p)

p ↔ q ≡ _________

r is a sufficient condition for s means " if r(x) then s(x)." r is a necessary condition for s means "if ~r(x) then ~s(x)"

r is a sufficient condition for s means... r is a necessary condition for s means ...

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verify the logical equivalence using Laws ∼(∼p ∧ q) ∧ (p ∨ q) ≡ p.