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Unformatted text preview: Mathematical Structures Horst R. Thieme Arizona State University, Supplementary Course Notes Spring 2007. c April 30, 2007 2 Chapter 1 Logic and Proof 1.1 Logical connectives 1.1.1 Example (logical equivalence). We consider the following statement A . A Today I take the car and do not put on a sweater, or I put on a sweater. Logically this seems to say the same as the following statement B . B Today I take the car, or I put on a sweater. 1.1.2 Definition (logical equivalence). Suppose A and B are composite statements formed from a given collection of statements p,q,r,s, ... using logical connectives. Then A and B are (logically) equivalent , if A and B have the same truth values for all possible truth values of the statements p,q,r,s,... . In order to prove that the statements A and B in Example 1.1.1 are equivalent we introduce c I take the car. s I put on a sweater. Then the composite statements A and B are given by ( c ∧ ∼ s ) ∨ s and c ∨ s . 3 4 CHAPTER 1. LOGIC AND PROOF c s ( c ∧ ∼ s ) ∨ s c ∨ s T T F F k T k k T k T F T T k T k k T k F T F F k T k k T k F F F T k F k k F k Double negation law Jim Kolbe (as chairman of the House commission that oversees the U.S. foreign aid budget): The lesson we have to learn from the tragic events of September 11 th is that we can’t not be involved, that we must continue to be involved. ∼ ( ∼ p ) is equivalent to p . In colloquial English, the use is sometimes opposite (and false). I cannot find no money in my wallet. Which means I really cannot find any money in my wallet (and I am upset about it) . 1.1.3 Theorem (De Morgan’s laws). Let p and q be statements. (a) ∼ ( p ∨ q ) is equivalent to ∼ p ∧ ∼ q . (b) ∼ ( p ∧ q ) is equivalent to ∼ p ∨ ∼ q . Proof. The following truth table shows that the two composite statements in (a) have the same truth values for all possible combinations. p q ∼ ( p ∧ q ) ∼ p ∨ ∼ q T T k F k T F k F k F T F k T k F F k T k T F T k T k F T k T k F F F k T k F T k T k T 1.1. LOGICAL CONNECTIVES 5 Associative Laws ( p ∧ q ) ∧ r is equivalent to p ∧ ( q ∧ r ). ( p ∨ q ) ∨ r is equivalent to p ∨ ( q ∨ r ). For the proof you need to consider 8 combinations of truth values. Commutative law p ∧ q and q ∧ p are logically equivalent. p ∨ q and q ∨ p are logically equivalent. Distributive laws p ∨ ( q ∧ r ) is equivalent to ( p ∨ q ) ∧ ( p ∨ r ). p ∧ ( q ∨ r ) is equivalent to ( p ∧ q ) ∨ ( q ∧ r ). ( p ∨ q ) ∧ r is equivalent to ( p ∧ r ) ∨ ( q ∧ r ). ( p ∧ q ) ∨ r is equivalent to ( p ∨ r ) ∧ ( q ∨ r ). Again the proof requires to consider 8 combinations of truth values. Contradiction A contradiction is a composite statement which is false for all possible combinations of truth values. For instance, p ∧ ∼ p is equivalent to F....
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This note was uploaded on 05/11/2008 for the course MAT 300 taught by Professor Thieme during the Spring '07 term at ASU.
 Spring '07
 thieme
 Logic

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