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10 - LECTURE 10 COUNTER-EXAMPLES AND CONTRADICTIONS(5.1-5.2...

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LECTURE 10: COUNTER-EXAMPLES AND CONTRADICTIONS ( § 5.1-5.2) Great spirits have always encountered violent opposition from mediocre minds. Albert Einstein (1879 - 1955) 1. Counter-examples We have often been dealing with proving universal statements like R : x S, P ( x ) . But what if R is false? How do we prove that? If R is false, then its negation is true. So we want to prove: R : x S, P ( x ) . We do this by finding some element x S where P ( x ) is false. This element x is called a counter- example . There may be more than one counter-example. Result. Show the following statement is false: For all x R , x 2 - x 6 = 0 . Proof. If x = 0 then x 2 - x = 0. Thus x = 0 is a counter-example. (So is x = 1.) ± We could fix the previous statement if we use a different domain: For all x R -{ 0 , 1 } , x 2 - x 6 = 0. Result. Show the following statement is false: If n is an even integer then n 2 - n is odd. What is the negation of this statement? It is: There exists some n Z with n an even integer and n 2 - n is even. Proof.
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10 - LECTURE 10 COUNTER-EXAMPLES AND CONTRADICTIONS(5.1-5.2...

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