2. (a) Compute and simplify f = b'c(a+b)'(ab'c + a)'(a'c + b) (b) Prove or disprove that in any Boolean algebra, B: [xa=ya and xa'=ya'] imply x = y, Va, x,y e B

9. Identify the Idempotent Law for AND and OR. AND: xy = yx and OR: x + y = y + x AND: xx¹ = 0 and OR: x + X' = 1 AND: 1x = x and OR: 0 + x = x AND: xx = x and OR: x + X = X

Q1 type answer only no need to explain

7. (a) Prove with Boolean algebra that (x + y) + (y · x') = x + y. Give a reason for each step in your proof. (b) Show the dual proof of part (a).

Source:https://www.mog.dog/files/SP2019/Sipser_Introduction.to.the.Theory.of.Computation.3E.pdf

Answer the following questions: a. Apply DeMorgan's Theorems to the following expression F = (ABC + DEF)' b. Apply DeMorgan's Theorems to the following expression F = ((A+B+C). D)' c. Use DeMorgan's Theorem in algebraic form to Prove F = A' + B' (A. B)' d. Use DeMorgan's Theorem in algebraic form to Prove F = A'. B' = (A + B)' e. Minimize the following Logic Expressions F, Y Algebraically and PROVE the function F to be equal to A'.C' + B, and the function Y to be equal to A+BC. A'.B'.C + A'.B.C'+A'.B.C+A.B.C'+A.B.C F (A, B, C) Y (A, B, C) (A+B) (A+C)

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

9. Given the following Boolean Function: F(A, B, C) = AB+B(Ā+C) Determine the canonical form for the SOP (sum of minterms) and POS (product of maxterms). Also, draw the truth tables showing the minterms and maxterms.

2. (a) Compute and simplify f = (xz+ (y+t)' + (xz)'y')' Encircle final answer! (b) Prove or disprove that in any Boolean algebra, B: (i) 0' = 1 (ii) 1' = 0