Question

The Boolean expression \(\overline {A + \overline B + C} + \overline {\overline A + \overline B + C} + \overline {A + \overline B + \overline C } + ABC\) reduces to:

1. A
2. B
3. C
4. A + B + C

Answer 1

Correct Answer - Option 2 : B

Concept:

De Morgans law:

The complement of the union of two sets is the intersection of their complements.

(x + y)’ = x’ . y’

The complement of the intersection of two sets is the union of their complements.

(x.y)’ = x’ + y’

Analysis:-

\(\overline {A + \bar B + C} + \overline {\bar A + \bar B + C} + \overline {A + \bar B + \bar C} + ABC\)

Applying Demorgan’s rule:-

\(\rightarrow \bar A \cdot B \cdot \bar C + AB\bar C + \bar ABC + ABC\)

\(\rightarrow \bar AB\left( {\bar C + C} \right) + AB\left( {\bar C + C} \right)\)

\(\rightarrow \bar AB + AB\)

\(\rightarrow B(A + \bar A)\)

→ B