Question

Simplify the following expressions to a minimum sum of products. (a) [(AB)' + C'D]' (b) [A + B(C' + D)]' (c) (A + B'))'(A + B)(C+ A)' Simplify each of the following expressions by applying only the stated laws or theorems once each. (a) Complementarity law: (A' + B' + C)(A' + B' + C)' (b) Absorption theorem: AB(C' + D) + B(C' + D) (c) Elimination theorem: AB + (C' + D)(AB)' (d) Distributive law then idempotent law: (A'BF + CD')(A'BF + CEG) (e) Elimination theorem: A'(B+C)(D'E + F)' + (D'E + F)

Answer 1

In order to simplify each expression, we can use various laws and theorems such as De Morgan's law, distributive law, double complement law, and idempotent law. By applying these laws and theorems step by step, we can simplify each expression to a minimum sum of products.

Step-by-step explanation:

(a) [(AB)' + C'D]':

Using De Morgan's law to distribute the complement over the addition and multiplication, we get:

[(AB)' + C'D]' = (AB)'C'D' = (A' + B')C'D'

Applying the distributive law, we get:

(A' + B')C'D' = A'C'D' + B'C'D'

This is the simplified expression.

(b) [A + B(C' + D)]':

Using the distributive law, we get:

[A + B(C' + D)]' = A'(C' + D)' + B'(C' + D)'

Applying De Morgan's law to distribute the complement over the addition, we get:

A'(C' + D)' + B'(C' + D)' = A'(C'D) + A'D' + B'C'D' + B'D'

This is the simplified expression.

(c) (A + B'))'(A + B)(C+ A)':

Using De Morgan's law to distribute the complement over the addition, we get:

(A + B'))'(A + B)(C+ A)' = (A'B')'(A + B)(C+ A)'

Applying De Morgan's law to distribute the complement over the addition, we get:

(A'B')'(A + B)(C+ A)' = (A'' + B'')'(A + B)(C+ A)'

Applying the double complement law, we get:

(A'' + B'')'(A + B)(C+ A)' = (A + B)(A + B)(C+ A)'

Applying the idempotent law and distributive law, we get:

(A + B)(A + B)(C+ A)' = (A + B)(C + A)'

This is the simplified expression.