Question

Simplification theorems (Boolean)

1. X+XY=X

2.X(X+Y)=X

3.(X+Y')Y=XY

Answer 1

(1)

X+XY =X

LHS

X+XY =X(1+Y)

Where 1+Y=1

$$X+XY=X(1+Y)=X\times1=X$$

$X+XY=X$

LHS=RHS

(2)

X(X+Y)=X

LHS

$X(X+Y)=X\times X+X\times Y$

XX=X

$X(X+Y)=X+XY$

$X(X+Y)=X(1+Y)$

Where 1+Y=1

$X(X+Y)=X\times1$

X(X+Y)=X

LHS=RHS

(3)

(X+Y')Y=XY

LHS

(X+Y')Y=XY+YY'

Where YY'=0

(X+Y')Y=XY+0

(X+Y')Y=XY

LHS =RHS