# XOR Gate & XNOR Gates: Truth Table, Boolean Expression

An exclusive-OR (XOR) gate is a digital logic gate that produces a true output (1) when exactly one of its inputs is true, and a false output (0) when both inputs are true or both inputs are false.

Truth Table

The following table shows the truth table for an XOR gate:

Symbol

The following is the symbol for an XOR gate:

Boolean Expression

The Boolean expression for an XOR gate is as follows:

``````A ⊕ B = (A + B) ⋅ (A' + B')
``````

where:

• A and B are the inputs to the XOR gate
• A’ and B’ are the complements of A and B, respectively
• The symbol ⊕ represents the XOR operation

Applications

XOR gates are used in a variety of digital circuits, including:

• Comparators
• Parity generators and checkers
• Encoders and decoders

XNOR Gate

An exclusive-NOR (XNOR) gate is a digital logic gate that produces a true output (1) when both of its inputs are true or both of its inputs are false, and a false output (0) when exactly one of its inputs is true.

Truth Table

The following table shows the truth table for an XNOR gate:

Symbol

The following is the symbol for an XNOR gate:

Boolean Expression

The Boolean expression for an XNOR gate is as follows:

``````A ≢ B = (A + B)'
``````

where:

• A and B are the inputs to the XNOR gate
• The symbol ≢ represents the XNOR operation

Applications

XNOR gates are used in a variety of digital circuits, including:

• Parity generators and checkers
• Encoders and decoders
• Code converters
• Arithmetic circuits

Example

One common example of the use of XOR and XNOR gates is in parity generators and checkers. Parity generators are used to add a parity bit to a digital signal to detect errors. Parity checkers are used to check the parity bit of a digital signal to detect errors.

The following is a diagram of a parity generator circuit:

The parity generator circuit has three inputs, A, B, and C, and one output, Parity. The Parity output is a 1 if the number of 1s in the inputs is even, and a 0 if the number of 1s in the inputs is odd.

The XOR gates in the parity generator circuit are used to calculate the Parity output. The Parity output is equal to a 1 if any of the inputs are equal to 1, and 0 otherwise.