By Smith D., Eggen M., Andre R.
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There are lots of bits and items of folklore in arithmetic which are handed down from consultant to pupil, or from collaborator to collaborator, yet that are too fuzzy and non-rigorous to be mentioned within the formal literature. usually, it used to be an issue of success and placement as to who realized such folklore arithmetic.
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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.