Idempotent law

Also called: idempotence, idempotent, idempotent laws, idempotency

The Boolean laws A + A = A and AA = A: repeating a variable or a term in an OR or an AND changes nothing.

The idempotent law says that repeating something has no effect:

Try both values: 0 + 0 = 0, 1 + 1 = 1, 0 · 0 = 0, 1 · 1 = 1. In every case the result equals A. There is no 2A or A² in Boolean algebra, because the only values are 0 and 1.

It works for whole terms too: = , and = A.

Used forwards, it deletes repeated terms after you multiply out brackets.

Used backwards, it is a powerful trick: you can write a term twice, X = , so that one term can be combined with two different partners. For example, in the term pairs with both of the others, giving . The same idea is why a minterm can be reused in several groups on a karnaugh map.

Worked examples

Example

Using a term twice

Simplify .

1.
Copy (idempotent): .
2.
= (combining on C).
3.
= (combining on B).
4.
Result: .

Example

Deleting a repeat

Multiplying out term by term gives .

1.
= A (idempotent).
2.
So the expression is .
3.
Absorption then removes and , leaving .

Common mistakes

Writing as 2A, or as A². Those come from ordinary algebra and do not exist here.
Thinking a term can only be used once when combining. You may copy any term as often as you like.

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Idempotent law is taught in Boolean Algebra.

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