Write the statement for the following switching circuit.
Simplify the statement.
Construct the switching circuit for the simplified form.
The corresponding Boolean expression for the given switching circuit is:
$ABC + ABC' + AB'C + A'BC$
This can be simplified as:
$AB \left(C + C'\right) + AB'C + A'BC$
$= AB + AB'C + A'BC$ $\;\;\;$ $\left[\text{Note: } C + C' = 1\right]$
$= B \left(A + A'C\right) + AB'C$
$= B \left(A + A'\right) \left(A + C\right) + AB'C$ $\;\;\; \left[\text{Distributive law: } a + bc = \left(a + b\right) \left(a + c\right)\right]$
$= B \left(A + C\right) + AB'C$ $\;\;\;$ $\left[\text{Note: } A + A' = 1\right]$
$= AB + BC + AB'C$
$= A \left(B + B'C\right) + BC$
$= A \left(B + B'\right) \left(B + C\right) + BC$ $\;\;\; \left[\text{By Distributive law}\right]$
$= A \left(B + C\right) + BC$ $\;\;\;$ $\left[\text{Note: } B + B' = 1\right]$
