Prove by simple induction on n that 2n > n
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- 10 Is A-(BU C) = (A-B) (A-C)? Prove it or disprove it.(b) Let n be an integer. Show that n if the sum of the digits of n is a multiple of 3,then n is a multiple of 3. You must do so by a direct proof. For this problem youmay assume that ∀a, b ∈ Z, ∀m ∈ Z+, (a ≡ x (mod m) ∧ b ≡ y (mod m)) ⇒ ab ≡ xymod m. *Solve by direct proof*7- Prove that (A+B)(A+C)=A+BC.
- need proofProblem 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.Show that the following argument is valid for any natural number n ∈N. p1 →p2 p2 →p3 ... pn−1 →pn ¬pn ∴¬(p1 ∨···∨pn−1................
- Prove that if x, y, and z are integers and x+y+s is odd, then at least one of x, y, and z is odd. Use the proof by cases to prove that there are no solutions in integers x and y to the equation 4x* +3y = 20 or find such integer solution.Suppose that the equation ax b .mod n/ is solvable (that is, d j b, whered D gcd.a; n/) and that x0 is any solution to this equation. Then, this equation has exactly d distinct solutions, modulo n, given by xi D x0 C i.n=d / fori D 0; 1; : : : ; d 1Given A = {1,2,3} and B={u,v}, determine. a. A X B b. B X B