Use a direct proof to show that if x is a real number such that (x – 1)2 = 0, then x3 –1 = 0.
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Q: proof that if x is an integer and x3 + 11 is odd, then x is even using a proof by contradiction.
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- A prime number is a natural number greater than 1 which is not a product of two smaller natural numbers. Prove or disprove: For every integer q, if q > 7, then q can be written as q = (a + b * c) such that the following properties hold a and b are prime numbers, c is an integer greater than 1.Note: For all integers k,n it is true that kn, k+n, and k-n are integers. An integer k is even if and only if there exists an integer r such that k=2r. An integer k is odd if and only if there exists an integer r such that k=2r+1. For every integer k it is true that if k is even then k is not odd. For every integer k it is true that if k is odd then k is not even. For every integer k it is true that if k is not even then k is odd. For every integer k it is true that if k is not odd then k is even. If P then Q means the same thing as P → Q (P implies Q). 3. Consider the argument form pvr .. p→r Is this argument form valid? Prove that your answer is correct. 4. Prove that for every integer d, if d³ is odd then d is odd.Prove: Let a, b, and c be integers. If (a - b) | c, then a | c. bk Suppose a, b, c are integers with (a b) | c. Then c = .). (² X . (a. b) for some integer k, so c = (alel X a, so a c.
- Prove: Let a, b, and c be integers. If Suppose a, b, c are integers with (a (a - b) | c, then a | c. b) | c. Then c = (a b) for some integer k, so c = ]a, · so a c.Prove: For all integers n, if n² is odd, then n is odd. Use a proof by contraposition, as in Lemma 1.1. Let n be an integer. Suppose that n is even, i.e., n = for some integer k. Then n² = is also even.Given A = {1,2,3} and B={u,v}, determine. a. A X B b. B X B
- - Q1: Prove that: + ((X.Y). (X.Z)) = X + Y.Z?proof that if x is an integer and x3 + 11 is odd, then x is even using a proof by contradiction.Given n is a positive integer, prove that n is odd if and only if 7n − 5 is even.You must use definitions of even (and odd, if needed). Do not use “even plus even is even” or similar rules of thumb.
- 6. Prove: For all integers n, if n² is odd, then n is odd. Use a proof by contraposition, as in Lemma 1.1.Let a be an integer such that a = 4 mod 12. Use the definition of mod to prove that a? = 4 mod 12.Prove the following statement by contraposition.For every integer x, if 5x2 – 2x + 1 is even, then x is odd.