Tautology

Also called: tautologies, contradiction, always true, always false, constant function

An expression that equals 1 for every input, such as A + A'. Its opposite, which is always 0 (like AA'), is called a contradiction.

A tautology is an expression that is 1 on every row of its truth table, whatever the inputs. The simplest is : one of A and is always 1. Its opposite, an expression that is 0 on every row, is a contradiction, such as .

A tautology simplifies all the way to the constant 1, and a contradiction to 0. In a circuit, the output could be replaced by a wire tied high or low.

They matter in simplification because they make terms vanish:

A term containing both X and , like , is a contradiction, so it is 0 and can be deleted from an SOP.
A bracket that is a tautology, like , is 1 and can be dropped from a product.

Checking: to show an expression is a tautology, show every row is 1, or simplify it to 1. To show it is not, find one row that gives 0.

A trap: looks like "something plus its complement", but it is not a tautology. It is 0 whenever A ≠ B. The true complement of is .


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

Worked examples

Example

A tautology in disguise

Show that is a tautology.

1.
De Morgan: = .
2.
So the expression is .
3.
By the complement law, that is 1 for every input.
4.
Check A = 1, B = 0: 0 + 0 + 1 = 1.

Example

Not a tautology

Is always 1?

1.
Try A = 1, B = 0: = 0 and = 0.
2.
The output is 0, so it is not a tautology.
3.
It is actually XNOR, which is 1 only when A = B.

Common mistakes

Assuming two terms with opposite-looking bars always OR to 1. Only an expression and its true complement do.
Writing = 0. A variable ORed with its complement is always 1.

Practice Tautology

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Tautology is taught in Boolean Algebra.

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