# Exclusive OR/X-OR gate(Inequality detector)

## Introduction

X-OR or Exclusive OR gate is a two input one output digital circuit. It gives a HIGH output when only one of its inputs is HIGH.

It gives a LOW output when both of its inputs are HIGH or LOW.

The output is high when the inputs are unequal so it is called inequality detector.

The symbol of X-OR operation is ⊕.

Three input X-OR gate doesn’t exist. When more than two inputs are to be X-ORed, a number of two inputs X-OR gates are used.

The output of X-OR gate becomes high when odd number of inputs are high. So it is an odd function.

#### Equivalent circuit of X-OR gate:

The expression of X-OR gate is X= A⊕B= $\overline{A}B+A\overline{B}$

So the equivalent circuit using AND,OR, INVERTER logic can be:

### X-OR gate as inverter:

X-OR gate can be used as an inverter. When we keep one input constant HIGH and vary other one it will act as an INVERTER.

In that case, when variable input is HIGH:

1⊕1=$\overline{1}.1+1.\overline{1}$

=0 + 0

=0

When variable input is LOW:

$1\oplus&space;0=\overline{1}.0+1.\overline{0}$

=0+1.1

=1

### X-OR gate ICs:

 IC Name Number of X-OR gates TTL IC 7486 4 CMOS IC 74C86 4 HIGH SPEED CMOS IC 74HC86 4

### Properties of X -OR gate:

1. $A\oplus&space;A=0$

Proof:

$A\oplus&space;A=&space;\overline{A}.A+A.\overline{A}$

= 0 + 0

= 0

2. $A\oplus&space;1=\overline{A}$

Proof:

$\overline{A}\oplus&space;1=&space;\overline{A}.1+A.\overline{1}$

= $\overline{A}+A.0$

$=\overline{A}$

3. $A\oplus&space;0=A$

Proof:

$A\oplus&space;0=\overline{A}.0+A.\overline{0}$

= 0 +A.1

= A

4. $AB\oplus&space;AC=A(B\oplus&space;C)$

Proof:

$AB\oplus&space;AC=\overline{AB}.AC+AB.\overline{AC}$

$=(\overline{A}+\overline{B}).AC+AB.(\overline{A}+\overline{C})$

$=A\overline{B}C+AB\overline{C}$

$=A(\overline{B}C+B\overline{C})$

$=&space;A(B\oplus&space;C)$

5. $A\oplus&space;\overline{A}=1$

Proof:

$A\oplus&space;\overline{A}=\overline{A}.\overline{A}+A.\overline{\overline{A}}$

$=\overline{A}+A.A$

$=\overline{A}+A$

= 1

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