Let f : B 3 → B where f(x, y, z) = (x + z)y. (a) Provide a truth table for the function. (b) Derive the canonical DNF for the function using the truth table. (c) Derive the canonical CNF for the function using the truth table.

1) The truth table of function fis given below left. Please complete its k-map in the right. XI X2 X3 X3

. Obtain the canonical SOP form of the following functions:                                    (i) y(A,B) = A+B (ii) f (A,B,C) = A+BC

2. Show that each of the following formulas are both satisfiable and falsifiable. (a) y Vxp(x, y). (b) Vr (p(x) V q(x)) → (Vxp(x)) v (Vxq(x)) (c) Vx (yp(x,y) →yq(x, y)).

2. Assume you have the following truth tables for functions F₁(x, y, z) and F₂(x, y, z): X y 0 0 0 0 1 1 1 1 oK NOH 0 0 0 1 1 1 0 0 0 1 1 0 1 1 0 F₁ --| 1 1 0 0 0 1 0 0 F2 ононононл 1 1 1 1 a) Express F₁ and F₂ in sum-of-products form b) Simplify each function

5. Let f:X→Y and g:Y→Z be one-to-one correspondences. (a) Prove that g◦fmust be a one-to-one correspondence. (b) Prove that (g◦f)−1=f−1◦g−1.

1. Given the following Minterm Expansion. Plot the terms on a suitable K-Map and then (a) find out the essential prime implicants(if any) and then (b) determine the minimum SOP for the function. You may need additional prime implicants. F(a, b, c) = Im(0, 1, 2, 4, 6)

1. Simplify the following Boolean functions, using four-variable maps: (a) F1(w,x,y,z) = E(1, 4, 5, 6, 12, 14, 15) (b) F2(A,B,C,D) = [(0,1,2,4,5,7,11,15) (c) F3(W,x,y,z) = II(0,1,4,5,6,7,8,9)

3. Use the Karnaugh-Map to simplify the following expressions F(A, B, C) = ABC + ABC + ABC + ABC + ABC F(A,B,C,D) = A + AB + ABC + BCD + ACD ii.

4. Karnaugh Maps Simplify parts a through d using Karnaugh maps a) F(x.y.)-Σ(0,2,4,5) b) F(x,y,z)=x'y'+yz+x'yz' c) F(w,x,y,z)=E(2,3,12,13,14,15) d) x'z+w'xy'+w(x'y+xy') e) Find minterms of xy+yz+xy'z by plotting it in a Karnaugh map. f) Find all the prime implicants of F (w, x, y, z)=E(0, 2, 4, 5, 6, 7, 8, 10, 13, 15) Determine which are essential. g) Simplify F (A, B, C, D)=E(1, 3, 4, 5, 10, 11, 12, 13, 14, 15) using essential prime implicants.