Redundant literal rule

Also called: redundant literal, dropping a barred letter, X + X'Y = X + Y, absorption variant, elimination theorem

The Boolean rule A + A'B = A + B (and its dual A(A' + B) = AB): the complemented copy of a lone variable can be dropped from another term.

This close relative of the absorption law removes a single barred letter rather than a whole term:

=
Dual: =

Why it works. If A = 1, both sides are 1. If A = 0, then = 1, so is just B, and both sides equal B. The never changes anything, so it can go.

Proof with the laws: = (OR over AND) = = .

How to spot it: a lone literal X in one term, and its complement inside another term. Delete the . X can be anything, including a barred letter: in , X = and = B, so the result is .

Do not confuse the two rules:

= A (absorption: the whole term goes).
= (this rule: only the barred letter goes).


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

Worked examples

Example

When the lone letter is barred

Simplify .

1.
The lone literal is . Its complement, C, appears in .
2.
Drop C from : the result is .
3.
Check B = 0, C = 1: the original is 0 + 0 = 0, and = 0 + 0 = 0. When C = 0, both are 1.

Example

The dual form

Simplify .

1.
Multiply out: = = .
2.
Or apply the dual rule directly: the inside the bracket is redundant.
3.
Result: .

Common mistakes

Writing = A. That treats it as absorption; the correct answer is .
Dropping the plain letter instead of the barred copy. Only the complemented copy of the lone literal goes.
Applying it when the lone term has more than one literal, such as . That is not this pattern.

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Boolean Algebra: Exam 2 · 1 question

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Redundant literal rule is taught in Boolean Algebra.

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