Absorption law

Also called: absorption, absorptive law, absorption laws, absorption theorem, covering theorem

The Boolean laws A + AB = A and A(A + B) = A: a term that contains all of another term adds nothing and can be deleted.

The absorption law removes a whole term:

= A
Dual: = A

Intuition. In , whenever is 1, A is already 1. So the longer term never changes the output, and the shorter one "absorbs" it.

Proof using earlier laws: = = = = A, using identity, distributive and null.

How to spot it in an SOP: if every literal of one term appears in another term, delete the longer one. In , contains all of , so the result is . The letters can stand for blocks: = .

Absorption has a close cousin, the redundant literal rule = , which removes a barred letter instead of a whole term. Mixing up the two is one of the most common errors in the topic.

		A
0	0	0	0	0
0	1	0	0	0
1	0	1	1	1
1	1	1	1	1

Worked examples

Example

Absorbing two terms at once

Simplify .

1.
B is a single letter, and both and contain B.
2.
So B absorbs both of them: .
3.
Nothing in contains B or the other way round, so stop.
4.
Result: . Check A = 1, B = 0, C = 1: the original is 0 + 1 + 0 + 0 = 1, and = 0 + 1 = 1.

Example

The POS form

Simplify .

1.
Let X = . The expression is X · (X + B).
2.
By the dual absorption law, = X.
3.
Result: .

Common mistakes

Writing = . Absorption deletes the whole term: the answer is A.
Absorbing when the terms don't nest. In neither term contains the other, so nothing is absorbed.
Keeping the longer term and deleting the shorter. It is always the term with more literals that disappears.

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Absorption law is taught in Boolean Algebra.

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