UPSC 2014 Maths Optional Paper 2 Q8b — Step-by-Step Solution

15 marks · Section B

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

Question

For any Boolean variables $x$ and $y$ , show that $x+xy=x$ .

Technique

Boolean-algebra absorption law; can prove via distributivity + $1+y=1$ , truth table, or logical reasoning.

Solution

This is the absorption law in Boolean algebra. Three equivalent proofs.

Proof 1 — Algebraic manipulation

x+xy=x\cdot 1+x\cdot y=x(1+y).

In Boolean algebra, $1+y=1$ for any $y$ (since OR with $1$ is always $1$ ). So

x+xy=x\cdot 1=x.

Proof 2 — Truth table

$x$	$y$	$xy$	$x+xy$
$0$	$0$	$0$	$0$
$0$	$1$	$0$	$0$
$1$	$0$	$0$	$1$
$1$	$1$	$1$	$1$

The last column matches the first column ( $x$ ). So $x+xy=x$ for all values of $x,y$ .

Proof 3 — Logical interpretation

$x+xy$ says ” $x$ is true, OR ( $x$ AND $y$ ) is true.” If $x$ is false: both disjuncts are false. If $x$ is true: the first disjunct is already true. So the whole expression equals $x$ .

Answer

\boxed{\;x+xy=x.\;}