Equivalent expressions

Also called: logically equivalent, logical equivalence, Boolean equivalence, equal expressions, equivalent

Two Boolean expressions are equivalent when they give the same output for every input, so they describe the same function.

Two expressions are equivalent (or equal) when they produce the same output on every row of the truth table. They are then two ways of writing one boolean function, and a circuit built from either does exactly the same job.

There are two ways to show equivalence:

Truth table (perfect induction): compare the output columns row by row. All must match.
Algebra (algebraic proof): rewrite one expression into the other using the laws of boolean algebra.

To show two expressions are not equivalent, one row where they differ is enough. That row is a counterexample.

Equivalence is what makes simplification safe. Every law is an equivalence, so each step keeps the function the same while the expression gets cheaper.

A quick self-check after any simplification: plug in a couple of rows, ideally ones where the terms you changed matter. It won't prove you are right, but it catches most slips.


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

Worked examples

Example

Equivalent by table

Are and equivalent?

1.
Both are 1 on rows 010, 011, 101 and 111, and 0 everywhere else (see the table above).
2.
All 8 rows match, so they are equivalent.
3.
Algebraically, this is the consensus theorem: is redundant.

Example

Not equivalent: one row is enough

Is equivalent to ?

1.
The second form has dropped the , so look for a row where that matters: A = 1.
2.
Try A = 1, B = 0, C = 1.
3.
= 0 + 0 = 0.
4.
= 0 + 1 = 1.
5.
They differ, so they are not equivalent.

Common mistakes

Concluding two expressions are equal because they agree on a few rows. Agreement on 7 of 8 rows is still not equivalence.
Thinking expressions must look alike to be equal. and look different but are equivalent.

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Equivalent expressions is taught in Boolean Algebra.

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