Section 3.4 Equivalent Statements

What You Will Learn Equivalent statements DeMorgan’s Law Variations of conditional statements

Equivalent Statements Two statements are equivalent, symbolized ⇔ or ≡, if both statements have exactly the same truth values in the answer columns of the truth tables.

Equivalent Statements If the answer columns are not identical, the statements are not equivalent. Sometimes the words logically equivalent are used in place of the word equivalent.

Example 1: Equivalent Statements Determine whether the following two statements are equivalent. p ⋀ (q ⋁ r) (p ⋀ q) ⋁ (p ⋀ r)

Example 1: Equivalent Statements Solution

Example 1: Equivalent Statements Solution Because the truth tables have the same answer (column 3 for both tables), the statements are equivalent. Therefore, we can write p ⋀ (q ⋁ r) ⇔ (p ⋀ q) ⋁ (p ⋀ r)

Example 3: Which Statements Are Logically Equivalent? Determine which statement is logically equivalent to “it is not true that the tire is both out of balance and flat.” a) if the tire is not flat, then the tire is not out of balance. b) the tire is not out of balance or the tire is not flat. c) the tire is not flat and the tire is not out of balance. d) if the tire is not out of balance, then the tire is not flat.

Example 3: Which Statements Are Logically Equivalent? Solution Let p: The tire is out of balance q: The tire is flat. “It is not true that the tire is both out of balance and flat.” may be written ~(p ⋀ q) “If the tire is not flat, then the tire is not out of balance.” may be written ~q → ~p

Example 3: Which Statements Are Logically Equivalent? Solution p: The tire is out of balance q: The tire is flat. “The tire is not out of balance or the tire is not flat.” may be written ~p ⋁ ~q “The tire is not flat and the tire is not out of balance.” may be written ~q ⋀ ~p

Example 3: Which Statements Are Logically Equivalent? Solution p: The tire is out of balance q: The tire is flat. “If the tire is not out of balance, then the tire is not flat.” may be written ~p → ~q Construct truth tables for all five statements.

Example 3: Which Statements Are Logically Equivalent? Solution ~(p ⋀ q) is logically equivalent to ~p ⋁ ~q

Example 3: Which Statements Are Logically Equivalent? Solution “It is not true that the tire is both out of balance and flat.” is logically equivalent to “The tire is not out of balance or the tire is not flat.”

De Morgan’s Laws

Example 5: Using De Morgan’s Laws to Write an Equivalent Statement Write a statement that is logically equivalent to “It is not true that tomatoes are poisonous or eating peppers cures the common cold.”

Example 5: Using De Morgan’s Laws to Write an Equivalent Statement Solution Let p: Tomatoes are poisonous. q: Eating peppers cures the common cold. “It is not true that tomatoes are poisonous or eating peppers cures the common cold.” may be written ~(p ⋁ q)

Example 5: Using De Morgan’s Laws to Write an Equivalent Statement Solution Using the second of DeMorgan’s laws, the equivalent of ~(p ⋁ q) is ~p ⋀ ~q Therefore, the equivalent statement is “Tomatoes are not poisonous and eating peppers does not cure the common cold.”

The Conditional Statement Written as a Disjunction To change a conditional statement into a disjunction, negate the antecedent, change the conditional symbol to a disjunction symbol, and keep the consequent the same.

The Disjunction Written as a Conditional Statement To change a disjunction statement to a conditional statement, negate the first statement, change the disjunction symbol to a conditional symbol, and keep the second statement the same.

Example 7: Rewriting a Disjunction as a Conditional Statement Write a conditional statement that is logically equivalent to “The Oregon Ducks will win or the Oregon State Beavers will lose.” Assume that the negation of winning is losing.

Example 7: Rewriting a Disjunction as a Conditional Statement Solution Let p: The Oregon Ducks will win. q: The Oregon State Beavers will win. “The Oregon Ducks will win or the Oregon State Beavers will lose.” may be written p ⋁ ~q

Example 7: Rewriting a Disjunction as a Conditional Statement Solution To write an equivalent conditional statement of p ⋁ ~q negate the first statement change the disjunction to conditional keep the second statement ~p → ~q “If the Oregon Ducks lose, then the Oregon State Beavers will lose.”

The Negation of the Conditional Statement Written as a Conjunction ~(p → q) ≡ p ⋀ ~q

Example 9: Write an Equivalent Statement Write a statement that is equivalent to “It is false that if you hang the picture then it will be crooked.”

Example 9: Write an Equivalent Statement Solution Let p: You hang the picture. q: The picture will be crooked. “It is false that if you hang the picture then it will be crooked.” may be written ~(p → q) which is ≡ p ⋀ ~q so, “You may hang the picture and the picture will not be crooked.”

Variations of the Conditional Statement The variations of conditional statements are the converse of the conditional, the inverse of the conditional, and the contrapositive of the conditional.

Variations of the Conditional Statement Name Symbolic Form Read Conditional p → q “If p, then q” Converse q → p “If q, then p” Inverse ~p → ~q “If not p, then not q” Contrapositive ~q → ~p “If not q, then not p”

Example 10: The Converse, Inverse, and Contrapositive For the conditional statement “If the song contains sitar music, then the song was written by George Harrison,” write the a) converse. b) inverse. c) contrapositive.

Example 10: The Converse, Inverse, and Contrapositive Solution Let p: The song contains sitar music. q: The song was written by George Harrison. “If the song contains sitar music, then the song was written by George Harrison.” may be written p → q

Example 10: The Converse, Inverse, and Contrapositive Solution Let p: The song contains sitar music. q: The song was written by George Harrison. p → q, the converse is q → p. “If the song was written by George Harrison, then the song contains sitar music.”

Example 10: The Converse, Inverse, and Contrapositive Solution Let p: The song contains sitar music. q: The song was written by George Harrison. p → q, the inverse is ~p → ~q. “If the song does not contain sitar music, then the song was not written by George Harrison.”

Example 10: The Converse, Inverse, and Contrapositive Solution Let p: The song contains sitar music. q: The song was written by George Harrison. p → q, the contrapositive is ~q → ~p. “If the song was not written by George Harrison, then the song does not contain sitar music.”