Question For each of the following, determine whether argument is valid. You may use a counterexample or equivalence transformations to justify your answer. (a) P→ q (b) (c) קר :: 79 P→r q→r ¬(pv q) :: r P→r q→r qV¬r :: 19

Q2/ Simplify using K- map for the following : Simplify the Boolean function F(A, B, C, D, E) = (0, 2, 4, 6, 9, 13, 21, 23, 25, 29, 31) %3D

If a, b, m ∈ Z+, ab mod m = [(a mod m)(b mod m)] mod m Show the equality of this expression.

Prove or disprove each of the following statements. To prove a statement, you should provide a formal proof that is based on the definitions of the order notations. To disprove a statement, you can either provide a counter example and explain it or provide a formal proof. All functions are positive functions. Ex.) f(n) (g(n)) ⇒ g(n) = O(f(n)) f(n) € (g(n)), for large values of n we have M₁g(n) ≤ f(n) ≤ M2g(n) for some M₁ and M₂. This means we have f(n) ≤ g(n) ≤ f(n), which shows g(n) = (f(n)). a) f(n) o(g(n)) and f(n) & w(g(n)) ⇒ f(n) = O(g(n)) b) f(n) = (h(n)) and h(n) = (g(n)) ⇒ f(n) ΕΘ(1) g(n) c) f(n) (g(n)) ⇒ 2f(n) = (29(n))

Use a truth table (either by hand or with a computer program) to prove the commutative laws for ^ and _. Use a truth table (either by hand or with a computer program) to prove the associative laws for ^ and _.

1. Simplify the given Boolean function in sum of product form using tabulation method. F (A,B,C,D)=E(1, 2, 3, 5, 7, 9, 10, 11, 13, 15) Determine the following: List of Prime Implicant Minterm with single X C. List of Essential Prime Implicant d. Minterm with no check mark Prime Implicant with no checkmark in its minterm Simplified Function in sum of product form a. b. e. f.

Write true if (p → r) ∧ (q → r) and (p ∨ q) → r are logically equivalent, otherwise false.

Simplify the following Boolean functions, using three-variable K-maps: F = A’B’ + BC + A’BC’

If A = {1, 2, 6} and B = {2, 3, 5}, then the union of A and B is