← 2016 Paper 2

UPSC 2016 Maths Optional Paper 2 Q8c — Step-by-Step Solution

15 marks · Section B

Boolean algebra · Numerical Analysis · asked 15× in 14 yrs · Read the full method →

Question

Let $A,B,C$ be Boolean variables, $\bar A$ denote complement of $A$, $A+B$ is an expression for $A$ OR $B$ and $A\cdot B$ is an expression for $A$ AND $B$. Then simplify the following expression and draw a block diagram of the simplified expression, using AND and OR gates.

Technique

Absorption $A\cdot(A+X)=A$ kills the three maxterms that contain a bare $A$; the remaining $A\cdot(\bar A+B+C)=A(B+C)$ by $A\bar A=0$ and distribution; confirmed by an 8-row truth table.

Solution

Setup. Call the expression

We simplify by absorption, then confirm against a truth table.

Step 1 — Absorb the maxterms containing $A$

The leading factor is $A$. Any other factor of the form $(A+\cdots)$ is absorbed by $A$ in a product, because $A\cdot(A+X)=A$:

So three of the four bracketed factors vanish, leaving only the one without a bare $A$:

Step 2 — Eliminate the complement

Answer