By Smith D., Eggen M., Andre R.

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All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. qxd 12 CHAPTER 1 4/22/10 1:42 AM Page 12 Logic and Proofs (a) (b) P ⇒ Q is equivalent to (∼Q) ⇒ (∼P) because the third column in the truth table is identical to the sixth column in the table. P ⇒ Q is not equivalent to Q ⇒ P because column 3 in the truth table differs from column 7 in rows 2 and 3. Ⅲ We have seen cases where a conditional sentence and its converse have the same truth value. 1(b) simply says that this need not always be the case—the truth values of P ⇒ Q cannot be inferred from its converse Q ⇒ P.

74, ∞ ) because with this universe the truth set for x2 > 3 is the same as the universe. DEFINITION For an open sentence P( x), the sentence (∀x) P (x) is read “For all x, P (x)” and is true iff the truth set of P(x) is the entire universe. The symbol ∀ is called the universal quantifier. Examples. For the universe of all real numbers, (∀x)(x + 2 > x) is true. (∀x)(x > 0 ∨ x = 0 ∨ x < 0) is true. That is, every real number is positive, zero or negative. (∀x)(x ≥ 3) is false because there are (many) real numbers x for which x ≥ 3 is false.

DEFINITION Let P and Q be propositions. The converse of P ⇒ Q is Q ⇒ P. The contrapositive of P ⇒ Q is (∼Q ) ⇒ (∼P). ” The converse is false, but the sentence and its contrapositive are true. ” In this example, all three sentences are true. The previous two examples suggest that the truth values of a conditional sentence and its contrapositive are related, but there seems to be little connection between the truth values of P ⇒ Q and its converse. We describe the relationships in the following theorem.