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Question: 29 please...

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love that for all x, a, b e R a. fx + a = b, thenx = b-a. b. if a 0 and ax b, then x bla. 23. Prove by Contradiction that there is no largest positive real number 24. Prove by Contradiction that there is no smallest positive real number. 25. Suppose that n e Z and n factors as n a.b, where a, b e Z and both are positive. Use by Contradiction to show that either an or bsm. 26. Use the previous Exercise to prove: If n is not a prime number (that is, n is composite), has a prime factor which is at most n. 27. Write the contrapositive of the statement in the previous Exercise. Use this to decide if I prime or composite. For Exercises (28) to (31): Use the technique of Proof by Contrapositive to prove the fo statements. You may use De Morgans Law to simplify the contrapositive, when applicabl 28. For all a, b e Z: if a b is even, then either a is even or b is even. 29. For all a, b e Z: if a + b is even, then either a and b are both odd or both even. 30. For all a E Z: a2 is even if and only if a is even. 31. For all x, y e R: ifx .y is irrational, then either a is irrational or b is irrational. Negating Statements with Quantifiers: A logical statement that begins with a qu negated as follows: not (Vr : p) is equivalent to: 3x : not (p). This should make se not true that all x possess property p, then at least one x does not possess property p. Similarly: not (Bx: p) is equivalent to: Vx: not (p). Thus, the negation of All of my friends are Democrats is One of my frien Democrat. Notice that None of my friends are Democrats is wrong. Similarly the negation of One of my brothers įs lef handed i, All Af 29 please
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