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- For the theorem: If n is an odd integer, then n + 2 is odd, give a) a direct proof. b) a proof by contraposition. c) a proof by contradiction. of this theorem.3. Prove that if n is an integer leaving remainder 1 after division by 4, then = n. Provide the type of your proof and all details.Prove the following statement using a proof by contradiction. 1. For every integer n, if n² is odd, then n is odd.
- Write a formal proof of the following statement. Be sure to copy this statement on your paper before beginning the proof. For all integers n, n is even if and only if 3n is even. Note: An "if-and-only-if" statement requires that you prove both implications: "if n is even then 3n is even" and "if 3n is even then n is even."Q: The sum of all minterms of a Boolean function of n variables is 1. Prove the above statement for n= 3. Suggest a procedure for a general proof.Let P(n) be the statement that a postage of n cents can be formed using just 4-cent and 5-cent stamps." Use strong mathematical induction to prove that P(n) is true for n ≥ 12. Answer the following questions to show a complete proof (Please remember that you must prove the statement using strong mathematical induction) Show that the statements P(12), P(13), P(14), and P(15) are true, completing the basis step of the proof. What is the inductive hypothesis of the proof? What do you need to prove in the inductive step? Complete the inductive step for k ≥ 15.
- A. Let m, n are integers. Use a proof by contraposition to prove that if m*n is even, then m is even or n is even. B. Let n be an integer. Use a proof by contradiction to prove that if 3n+2 is even, then n is even.6. Prove: For all integers n, if n² is odd, then n is odd. Use a proof by contraposition, as in Lemma 1.1.4. Let P be the statement "If n´ is an odd integer, then n is odd integer." (a). What is the contraposition of P? (b). Prove P by contraposition.
- Problem 2: Consider the set of integers. 1. Use a direct proof to show that the sum of two odd integers is even. 2. Use a direct proof to show that the product of two odd integers is odd. Prove that if n is an integer and n³ + 5 is odd, then n is even using a) A proof by contraposition. 3. b) A proof by contradiction. 4. Prove or disprove that if m and n are integers such that mn = 1, then either m = 1 and n = 1, or else m = -1 and n = -1. 5. Prove that these four statements about the integer n are equivalent: a) n² is odd. b) 1 - n is even. c) n³ is odd. d) n² + 1 is even. 6. Prove that if m and n are integers and mn is even, then m is even or n is even.Determine the validity of the ff. proof: Theorem: The difference between any odd integer and any even integer is odd. "Proof: Suppose n is any odd integer, and m is any even integer. By definition of odd, n = 2k+1 where k is an integer, and by definition of even, m = 2k where k is an integer. Then n- m = (2k+1)– 2k = 1, and 1 is odd. Therefore, the difference between any odd integer and any even integer is odd." O Insufficient information to find the answer Yes, it is valid O Maybe O Not validProve the following using a direct proof: The sum of the squares of 4 consecutive integers is an even integer.