XOR gate

Also called: exclusive OR, exclusive-OR gate, EOR gate, EXOR gate, XOR logic gate

A logic gate whose output is 1 when its two inputs are different and 0 when they match. Written A ⊕ B, it equals A'B + AB'.

An XOR gate (exclusive OR) outputs 1 when its inputs are different: one is 1 and the other is 0. When both are 0 or both are 1, it outputs 0. It is written , shown as A ⊕ B, and typed with ^.

Compare it with OR. The two agree in rows 00, 01 and 10. They differ only in row 11, where OR gives 1 and XOR gives 0. "Exclusive" means one or the other, but not both.

Using only AND, OR and NOT, XOR is : one term picks out row 01, the other row 10.

Why XOR shows up everywhere:

Adding bits. The sum bit of 1-bit addition is XOR (0, 1, 1, 0), so XOR sits at the heart of every half adder and full adder.
Controlled inversion. = A and = , so one input can decide whether the other is flipped. See xor identities.
Parity. A multi-input XOR outputs 1 when an odd number of inputs are 1, which is how parity bits are made.
Comparing. XOR is 1 exactly where two bits disagree.

The symbol is the OR shape with an extra curved line behind the inputs.

Worked examples

Example

Deriving XOR's expression from its table

XOR is 1 in rows 01 and 10 only. Write it with AND, OR and NOT.


0	0	0	0
0	1	1	1
1	0	1	1
1	1	0	0

1.
Row 01 needs A = 0 and B = 1: the term .
2.
Row 10 needs A = 1 and B = 0: the term .
3.
The output is 1 when either term is 1, so OR them: .
4.
Check row 11: = 0 and = 0, so the output is 0, as XOR requires.

Example

A 4-input XOR

Find for A = 0, B = 1, C = 1, D = 1.

1.
Count the 1s: B, C and D, so three.
2.
Three is odd, so the output is 1.
3.
Chaining agrees: 0 ⊕ 1 = 1, then 1 ⊕ 1 = 0, then 0 ⊕ 1 = 1.

Common mistakes

Giving 1 for row 11. That's OR. XOR is 0 when both inputs are 1.
Thinking a 3-input XOR is 1 only when exactly one input is 1. It is 1 for any odd count, so 111 gives 1 too.
Mixing XOR with AND and OR without brackets. Write rather than relying on a precedence rule your reader may not share.

Practice XOR gate

Interactive questions with instant feedback and a worked solution for every wrong answer.

Boolean Algebra lesson

Fill in the output column for F = .

Logic Gates lesson

And 15 more in the Logic Gates lesson.

Combinational Logic lesson full course

Adders and ALUs lesson full course

And 6 more in the Adders and ALUs lesson.

Practice exams

Learn it step by step

XOR gate is taught in Logic Gates.

Logic Gates →

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