Logical Equivalence of Propositions Dr. Yasir Ali

Two statement forms are called logically equivalent if, and only if, they have identical truth values for each possible substitution of statements for their statement variables. The logical equivalence of statement forms p and q is denoted by writing p ≡ q.

Double Negation: Construct a truth table to show that the negation of the negation of a statement is logically equivalent to the statement. p ≡ ￢(￢p) p ￢p ￢(￢p) T F

Representation of If-Then As Or p →q ≡ ￢ p ∨ q p q ￢p p →q ￢p v q T F T T F F

￢p v q = I won’t catch the 8:05 bus or I am on time. Rewrite the following statement in if-then form. I am on time for work if I catch the 8:05 bus. Let p = I catch the 8:05 bus. q = I am on time. ￢p v q = I won’t catch the 8:05 bus or I am on time. Which can be re written as: I am on time or I won’t catch the 8:05 bus. You get to work hard or you will fail.

D’ Morgan Laws 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. ￢(p ∨ q) ≡ ￢ p ∧ ￢ q.

The connector is loose or the machine is unplugged. The dollar is at an all-time high and the stock market is at a record low. Assume x is a particular real number and use De Morgan’s laws to write negations for the statements: −2 < x < 7 x ≤ −1 or x > 1

Tautology and Contradiction A tautology is a statement form that is always true regardless of the truth values of the individual statements substituted for its statement variables. A statement whose form is a tautology is a tautological statement. A contradiction is a statement form that is always false regardless of the truth values of the individual statements substituted for its statement variables. A statement whose form is a contradiction is a contradictory statement.

Tautology and Contradiction The statement form p ∨ ￢ p is a tautology and that the statement form p ∧ ￢ p is a contradiction. If t is a tautology and c is a contradiction, show that p ∧ t ≡ p and p ∧ c ≡ c p ￢ p p ∨ ￢ p p ∧ ￢ p T F p t c p ∧ t p ∧ c T F

Some Important Equivalences Name p ∧ t ≡ p p ∨ c ≡ p Identity laws p ∨ t ≡ t p ∧ c ≡ c Domination laws p ∨ p ≡ p p ∧ p ≡ p Idempotent laws ￢(￢p) ≡ p Double negation law p ∨ q ≡ q ∨ p p ∧ q ≡ q ∧ p Commutative laws (p ∨ q) ∨ r ≡ p ∨ (q ∨ r) (p ∧ q) ∧ r ≡ p ∧ (q ∧ r) Associative laws p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r) p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r) Distributive laws ￢(p ∧ q) ≡ ￢p ∨￢q ￢(p ∨ q) ≡ ￢p ∧￢q De Morgan’s laws p ∨ (p ∧ q) ≡ p p ∧ (p ∨ q) ≡ p Absorption laws p ∨￢p ≡ t p ∧￢p ≡ c Negation laws

Verify the logical equivalence ∼(∼p ∧ q) ∧ (p ∨ q) ≡ p. ∼(∼p ∧ q) ∧ (p ∨ q) ≡ (∼(∼p)∨ ∼q) ∧ (p ∨ q) by De Morgan’s laws by the double negative law ≡ (p∨ ∼q) ∧ (p ∨ q) by the distributive law ≡ p ∨ (∼q ∧ q) by the negation law ≡ p ∨ c by the identity law. ≡ p

Show that (p ∧ q) → (p ∨ q) is a tautology.

Negation of Conditional Statements The negation of “if p then q” is logically equivalent to “p and not q.” ￢(p →q) ≡ p ∧ ￢ q ￢(p →q) ≡ ￢(￢ p ∨ q) ≡ ￢(￢ p) ∧ (￢ q) by De Morgan’s laws ≡ p ∧ ￢ q by the double negative law.

It is tempting to write the negation of an if-then statement as another if-then statement. Please resist that temptation! Exercise: If P is a square, then P is a rectangle. If x is nonnegative, then x is positive or x is 0. If today is New Year’s Eve, then tomorrow is January.

A conditional statement is logically equivalent to its contrapositive.

Exercises from text book Page 34-35 questions 1-33