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- Determine which of the following “proofs” are correct and which are incorrect. If a proof is correct, indicate the type and if a proof is incorrect, indicate why it is incorrect.Theorem: If a and b are even integers then a – b is an even integer. letter H and IProve that if p> 3 is a prime number, then 24 | (p2 - 1).Given that n is an integer, it follows that 2n is an even integer. O a. Inductive, strong. b. Deductive, invalid. c. Deductive, sound. O d. Inductive, weak. Oe. Deductive, valid.
- 13. Let S be the set of 7-digit numbers (neN: 106Given a positive integer n, let a1,n ... a2,n–1 a2,n ... a3,n–2 аз,п—1 a3,n .. An = аn-1,2 ап-1,n-2 аn-1,n-1 An-1,n ... an,1 An.2 An,n-2 An,n-1 An,n ... whose aij entries with i+j4. Do a direct proof using domain knowledge: An odd integer minus an even integer is odd.Prove the following proposition using forward-backward method, "If n is an even integer, then (n² + 2) is divisible by 2". Provide both the analysis of proof and the condensed proof.3. Show that if n is an integer and n3 + 5 is odd, then n is even using a) a proof by contraposition. b) a proof by contradiction.n > 1 is an integer. Which of the following statements is false? Select one: a. If 10" – 1 is prime then n is prime. b. If n is even then 2" + 1 is prime. c. If 2" – 1 is prime then n is prime. d. If n is composite then 10" – 1 is composite. e. If n is composite then 2" – 1 is composite.How many four-digit even numbers are possible if the leftmost digit cannot be zero? Repitition is allowes.prove the theorem "If n is odd, then n+1 is even" in the following ways: 1) Direct proof (Assume p, show q). 2) Proof by contraposition (Assume ~q, show ~p). 3) Proof by contradiction (Assume p and "q, then seek a contradiction).1. Prove by Contrapositive: "If n2 is odd, then n is odd. 2. Prove by mathematical induction: (ab)" = a" b" , for all integer n > 1SEE MORE QUESTIONSRecommended textbooks for youCalculus: Early TranscendentalsCalculusISBN:9781285741550Author:James StewartPublisher:Cengage Learning
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