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Unformatted text preview: MAT 300 Spring 2009 Chad Awtrey SCHOOL OF MATHEMATICAL AND STATISTICAL SCIENCES Section 1 — Logical Connectives Sentential Connectives EXERCIES Statements To understand mathematics and mathematical arguments, it is necessary to have a solid understanding of logic and the way in which known facts can combined to prove new facts! Definition A STATEMENT is any sentence that can be classified as true or false. A statement’s TRUTH VALUE is TRUE if the statement is true and FALSE if the statement is false. For a sentence to be a statement, it is not necessary that we know whether it is true or false, but it must clearly be the case that it is one or the other. MAT 300 Awtrey MATHEMATICS AND STATISTICS 2 / 35 Section 1 — Logical Connectives Sentential Connectives EXERCIES Statements Example Consider the following sentences. Which ones are STATEMENTS? If the sentence is a statement, what is its truth value? (a) Two plus two equals four. (b) Every continuous function is differentiable. (c) If x = 2, then x 2 5 x + 6 = 0. (d) A circle is the only convex set in the plane that has the same width in every direction. (e) Every even number greater than 2 is the sum of two primes. ( Goldbach’s Conjecture ) (f) My wife is the best cook. MAT 300 Awtrey MATHEMATICS AND STATISTICS 3 / 35 Section 1 — Logical Connectives Sentential Connectives EXERCIES Statements Answers: (a)–(e) are statements; but (f) is not a statement since the notion “best cook” means different things to different people. It should be noted that I believe (f) is true!! MAT 300 Awtrey MATHEMATICS AND STATISTICS 4 / 35 Section 1 — Logical Connectives Sentential Connectives EXERCIES Statements Practice Which of the sentences are STATEMENTS? What are their truth values? (a) If x is a real number, then x 2 . (b) Seven is a prime number. (c) Seven is an even number. (d) 7919 is a prime number. (e) This sentence is false. MAT 300 Awtrey MATHEMATICS AND STATISTICS 5 / 35 Section 1 — Logical Connectives Sentential Connectives EXERCIES Statements Answers: (a) statement true (b) statement true (c) statement false (d) statement true (it’s the 1000th prime number) (e) not a statement MAT 300 Awtrey MATHEMATICS AND STATISTICS 6 / 35 Section 1 — Logical Connectives Sentential Connectives EXERCIES Sentential Connectives not, and, or, if/then, if and only if Consider the statement It is raining Let’s call this statement p . We usually write statements in this manner: p : It is raining with a colon between the statement and its representing letter. We can form the negation of statement p , in symbols we write p , and we read this as “ not p ”. p : It is not raining MAT 300 Awtrey MATHEMATICS AND STATISTICS 7 / 35 Section 1 — Logical Connectives Sentential Connectives EXERCIES Negation not Definition If p is any statement, the negation of p (written p and read not p ) is the logical opposite of p . When p is true, then p is false. When p is false, then p is true....
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This note was uploaded on 02/22/2011 for the course MAT 300 taught by Professor Thieme during the Spring '07 term at ASU.
 Spring '07
 thieme
 Math, Logic

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