Question

Simplify the following expressions to a minimum sum of products. Only individual variables should be complemented. (a) [(XY") + (X' + Y) Z] (b) (X + (Y' (Z + W))) (c) ((A' + B') + (A'B'C' + CD)' (d) (A + B)CD + (A + B)

Answer 1

To simplify the given expressions to a minimum sum of products, apply the Boolean algebra rules. Distribute the operations and apply De Morgan's law to complement variables inside parentheses. Then, combine like terms to get the minimum sum of products.

Step-by-step explanation:

To simplify the given expressions to a minimum sum of products, we need to apply the Boolean algebra rules. Here are the steps for each expression:

(a)

1. Distribute the operation over the parentheses: (XY') + (X' + Y)Z.
2. Apply De Morgan's law to complement the variables inside parentheses: (XY') + (X'Z' + YZ).
3. Distribute again: XY' + X'Z' + YZ.
4. Combine the terms to get the minimum sum of products: XY' + X'Z' + YZ.

(b)

1. Distribute the operation over the parentheses: X + (Y'(Z + W)).
2. Apply De Morgan's law to complement the variable inside parentheses: X + (Y'Z'W' + Y'W').
3. Distribute again: X + Y'Z'W' + Y'W'.
4. Combine the terms to get the minimum sum of products: X + Y'Z'W' + Y'W'.

(c)

1. Distribute the operation over the parentheses: (A' + B') + (A'B'C' + CD)'
2. Apply De Morgan's law to complement the variables inside parentheses: (A' + B') + (A'B'C' + C' + D')
3. Distribute again: A' + B' + A'B'C' + C' + D'
4. Combine the terms to get the minimum sum of products: A' + B' + A'B'C' + C' + D'

(d)

1. Combine like terms: (A + B)CD + (A + B)
2. Apply the distributive law: ACD + BCD + A + B
3. Combine again: (ACD + A) + (BCD + B)
4. Combine the terms to get the minimum sum of products: ACD + A + BCD + B