This assignment due Saturday, 10/10 12pm EST. There are eight questions. Its mainly about Direct Proof and Proof by Contrapositive. More on Direct Proof and Proof by Contrapositive. Existence and Proof by Contradiction. Here’s the instructions.Prove the following results and theorems. Write out the statment of the theorem or result before proving it. Write your proof in complete, grammatical sentences, and in easily legible handwriting.
Result 1. Let a and c be odd integers. Then ab + bc is even for all integers b. Result 2. For all integers x, if 7x + 4 is even, then 3x − 11 is odd. (Hint: use a lemma to prove
this result.) Result 3. For any integer n, (n + 1)2 − 1 is even if and only if n is even. Theorem 4. For every two sets A, B, A ∩ B = A ∪ B. Theorem 5. For any four sets A, B, C, D, (A × B) ∩ (C × D) = (A ∩ C) × (B × D). Theorem 6. There do not exist positive integers m and n for which m2 − n
2 = 1. Theorem 7. There are infinitely many positive integers n for which √
n is irrational. (Hint:
Let n = 2m where m is an odd, positive integer, and show that √
n is irrational. Then, since
there are infinitely many odd positive integers m, there are infinitely many n = 2m for which
n is irrational. Theorem 8. There exist distinct irrational numbers a and b such that a
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