Give a carefully worded proof for the following statements using either direct or contrapositive proof. 1) 2) If a is an integer, then 2 | (a² +3a+ 4). If two integers have opposite parity, then their sum is odd.
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- Determine whether the following statements are logically equivalent or not. Show your work to clearly indicate your answer. p→¬qand¬(p→q)Show that the following statements are equivalent, where n is an integer greater than or equal to 2. Feel free to consider any of the following pairs. 1. “n is even” and “n – 1 is odd”2. “n is even” and “n2 is even”3. “n – 1 is odd” and “n2 is even” with Step By step explanation pleaseFor each of the statements below, determine for yourself if the statement is true orfalse. If it is true, write a direct proof. If it is false, use a counterexample to show that it is false(a) For all integers m and n, if mn ≡ 0 (mod 8), then m ≡ 0 (mod 8) orn ≡ 0 (mod 8).(b) For all integers a, if a ≡ 7 (mod 10), then a^2 ≡ 9 (mod 10).
- write the formal proof of the following statement: If m and n are odd integers and p is an even integer, then mn+p is an odd integer2. Prove that if x and y are real numbers, then max(x, y) + min(x, y) = x + y.[Hint: Use a proof by cases, with the two cases corresponding to x > y and x < y,respectively.]Prove that the following logic statement is logically true using proof by contradiction: (p→q)∧(q→r)→(p→r)
- Prove that the following logic statement is logically true. Please use proof by contradiction. (p→q)→(¬q→ ¬p)Use Conditional Proofs (CP) to show that each of the following symbolic arguments are valid. Commas mark the breaks between premises. P→(Q→R) ∴ Q→(P→R) P→Q ∴ P→(Q v R) (B∙D)→(C∙R), D ∴ B→R (Z∙~W)→(X∙U), Z→(W→Y) ∴ Z→(~Y→U) (E∨F)→G, A→[(B v C)→(D∙E)] ∴ A→(C→G)Consider the following argument ∀x, N(x) → J(x) ∃x, N(x) ∧ A(x) ∀x, A(x) → E(x) ∴ ∃x, J(x) ∧ E(x) a. Prove that the given argument is valid b. Provide an example whose symbolic translation is the argument above
- 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."1. In transforming a formula into an equivalent CNF, you can use the absorption laws to eliminate conjunctions within disjuntions, i.e. expressions such as p ˅ (q ˄ r) and (p ˄ q) ˅ r. True False 2. Let p, q, and r be propositional variables. Which of the following expressions would NOT be formulas in conjunctive normal form? (p ˄ ¬q) ˅ (¬r ˄ q ˄ p) ¬q p ˅ q ˅ ¬p p ˄ ¬p 3. Consider the propositional logic formula (p ˄ q) ˅ (r ˅ ¬s) From the options below, which one is the equivalent CNF? To determine the correct answer, transform the formula above into CNF using the steps learnt in this module. (p ˅ r ˅ s) ˄ (q ˅ r ˅ ¬s) (¬p ∨ r ∨ s) ∧ (¬q ∨ ¬r ∨ ¬s) (p ˅ r ˅ ¬s) ˄ (q ˅ r ˅ s) (p ˅ r ˅ ¬s) ˄ (q ˅ r ˅ ¬s)…Determine the truth value of each of the following statements, explain the reasons in short. a) if the universe of discourse for each variable consists of all real numbers:• ∀x∃y(y = x + 2)• ∀x∀y((x − y)2 = x2 − y2b) f : R → R f(x) = 2x2 + 1 is bijective funtion.c) (nlogn + n2)(n3 + 2) is O(n5)d) (n! + n2)(n3 + log(n2 + 1) is O(nn)