#### Transcript of Fatin, Favad, and Daniyal Present:

**Boolean Algebra and Logic**

Q.7 Identify the law of Boolean algebra upon which each of the following equalities is bound

(a) A¬B + CD + A¬CD + B = B + A¬B +A¬CD + CD

(b) AB¬CD + ¬(ABC) = D¬CBA + ¬(CBA)

(c) AB(CD+E¬F+GH) = ABCD + ABE¬F + ABGH

Ans.

(a) Commutative law of addition

(b) Commutative law of multiplication

(c) Distributive law

Q1. Using Boolean notation, write an expression that is 1 whenever one or more of its variables (A, B, C, and D) are 1s.

Ans. X = A + B + C + D

Q.9 Apply DeMorgan's theorems to each expression:

(a) ¬(A+¬B) (b) ¬(¬AB) (c) ¬(A+B+C) (d) ¬(ABC)

(e) ¬(A(B+C)) (f) ¬(AB) + ¬(CD)

(g) ¬(AB + CD) (h) ¬((A+¬B)(¬C+D))

(a) ¬(A+¬B) = ¬A¬(¬B) = ¬AB

(b) ¬(¬AB) = ¬(¬A) + ¬B = A + ¬B

(c) ¬(A+B+C) = ¬A¬B¬C

(d) ¬(ABC) = ¬A + ¬B + ¬C

(e) ¬(A(B+C)) = ¬A + ¬(B+C) = ¬A + ¬B¬C

(f) ¬(AB) + ¬(CD) = ¬A + ¬B + ¬C + ¬D

(g) ¬(AB + CD) = ¬(AB)¬(CD) = (¬A+¬B)(¬C+¬D)

(h) ¬((A+¬B)(¬C+D)) = ¬(A+¬B) + ¬(¬C+D) = ¬AB + C¬D

Fatin, Favad, and Daniyal Present:

Q. 12 Write the Boolean Expression for each of the following logic gates

(a) (b) (c) (d)

Ans.

(a) AB = Q

(b) A + B = Y

(c) ¬A = Q

(d) A + B + C = Output

Q.19 Using Boolean algebra techniques simplify the following expressions as much as possible:

(a) A(A+B) (b) A(¬A + AB) (c) BC + ¬BC (d) A(A + ¬AB) (e) A¬BC + ¬ABC + ¬A¬BC

Ans.

(a) A(A+B) = AA + AB = A + AB = A(1 + B) = A

(b) A(¬A + AB) = A¬A + AAB = 0 + AB = AB

(c) BC + ¬BC = C (B + ¬B) = C(1) = C

(d) A(A + ¬AB) = AA + A¬AB = A +(0)B = A

(e) A¬BC + ¬ABC + ¬A¬BC = A¬BC + ¬AC(B + ¬B) = A¬BC + ¬AC = C(¬A + A¬B) = C(¬A + ¬B) = ¬AC + ¬BC

Q.23 Convert the following expressions to sum of products form:

(a) (A+B)(C+¬B) (b) (A + ¬BC)C (c) (A+C)(AB+AC)

Ans.

(a) (A+B)(C+¬B) = AC + BC + A¬B + B¬B = AC + A¬B + BC

(b) (A + ¬BC)C = AC + ¬BCC = AC + ¬BC

(c) (A+C)(AB+AC) = AAB + AAC + ABC + ACC = AB + AC + ABC + ACC = (AB + AC)(1 + C) = AB + AC

Q.31 Develop a truth table for each of the following standard SOP expressions.

(a) A¬BC + ¬AB¬C + ABC (b) ¬X¬Y¬Z + ¬X¬YZ + XY¬Z + X¬YZ + ¬XYZ

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