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UPSC 2017 Maths Optional Paper 2 Q5c — Step-by-Step Solution

10 marks · Section B

Boolean algebra · Numerical Analysis · asked 15× in 13 yrs · Read the full method →

Question

Write the Boolean expression $z(y+z)(x+y+z)$ in its simplest form using Boolean postulate rules. Mention the rules used during simplification. Verify your result by constructing the truth table for the given expression and for its simplest form.

Technique

Repeated absorption $z\cdot(z+\text{anything})=z$; the factor $z$ outside absorbs both sum factors that already contain $z$.

Solution

Step 1 — Absorb $(y+z)$

By the absorption law $z(z+y)=z$ (equivalently $z\cdot(y+z)=z$, since $z+yz...$ — directly: $z(y+z)=zy+zz=zy+z=z(1+y)=z$):

Rules used: distributive ($z(y+z)=zy+zz$), idempotent ($zz=z$), then $z+zy=z(1+y)=z$ via identity $1+y=1$ and absorption.

So the expression becomes

Step 2 — Absorb $(x+y+z)$

Again by absorption, $z(x+y+z)=z$:

using distributive, idempotent ($zz=z$), and null/identity law $x+y+1=1$.

Step 3 — Simplest form

Answer