Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 3.4 Equivalent Statements

Copyright 2013, 2010, 2007, Pearson, Education, Inc. What You Will Learn Equivalent statements DeMorgan’s Law Variations of conditional statements 3.4-2

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 1: Equivalent Statements Determine whether the following two statements are equivalent. p ⋀ (q ⋁ r) (p ⋀ q) ⋁ (p ⋀ r) 3.4-5

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 1: Equivalent Statements Solution 3.4-6

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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) 3.4-7

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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 3.4-9

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 3: Which Statements Are Logically Equivalent? Solution ~(p ⋀ q) is logically equivalent to ~p ⋁ ~q

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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.”

Copyright 2013, 2010, 2007, Pearson, Education, Inc. De Morgan’s Laws

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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.”

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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)

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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.”

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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 Conditional Statement Written as a Disjunction

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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. The Disjunction Written as a Conditional Statement

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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.” Example 7: Rewriting a Disjunction as a Conditional Statement

Copyright 2013, 2010, 2007, Pearson, Education, Inc. The Negation of the Conditional Statement Written as a Conjunction ~(p → q) ≡ p ⋀ ~q

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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.”

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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.”

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Variations of the Conditional Statement NameSymbolic Form Read Conditionalp → qp → q“If p, then q” Converseq → pq → p“If q, then p” Inverse~p → ~q“If not p, then not q” Contrapositive~q → ~p“If not q, then not p”

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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 → qp → q

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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.”

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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.”

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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.”