Complement law

Also called: complementation law, inverse law, complement laws, complementarity law

The Boolean laws A + A' = 1 and AA' = 0: a variable ORed with its complement is always 1, and ANDed with it is always 0.

The complement law pairs a value with its opposite:

A and are always different, so exactly one of them is 1. Their OR is therefore always 1, and their AND is always 0.

The letter can stand for a whole expression. = 1 and = 0, because any expression is just some 0 or 1, and its complement is the other value.

This is the law that makes variables disappear during simplification. Every time two terms differ only in one letter's bar, you factor and use it:

= = = A.

That pattern is so common it has its own name, combining terms.

Careful: the law needs the same thing plain and complemented. is not 1, because is not the complement of .


0	1	0
1	1	0

Worked examples

Example

Deleting an impossible term

Simplify .

1.
= 0 (complement law).
2.
The expression is = (identity).
3.
C absorbs (absorption).
4.
Result: C.

Example

Checking a false claim

A student says = 1 because the two terms are complements. Test A = 1, B = 0.

1.
= 1 · 0 = 0.
2.
= 0 · 1 = 0.
3.
The sum is 0, not 1, so the claim is false.
4.
The true complement of is , by De Morgan's laws.

Common mistakes

Writing = 0 or = 1. OR of opposites is 1; AND of opposites is 0.
Treating as the complement of . The complement of a product needs De Morgan: .
Missing the law when the letter is a whole expression, such as = 0.

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Complement law is taught in Boolean Algebra.

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