Counterexample

Also called: counter-example, counterexamples, disproof by example

One input combination where two expressions give different outputs. A single counterexample proves they are not equal.

A counterexample is a specific input that breaks a claim. For a claimed identity between two expressions, it is a row where the two sides disagree.

The logic is lopsided:

To prove two expressions are equivalent, you must cover every row, or use the laws.
To disprove it, one row is enough.

That makes counterexamples the fastest tool for checking your own work. If you suspect a step, try an input that makes the changed part matter.

How to find one quickly: look at what differs between the two sides and choose values that make that part decide the output. For versus , the difference only shows when A = 0 (otherwise both are 1), and then B = 1 makes true but false.

Counterexamples also explain the common myths. Cancelling fails because A = 1, B = 0, C = 1 makes = while B ≠ C.

Worked examples

Example

Disproving a false De Morgan

A student claims = . Find a counterexample.

1.
The two sides can only differ when exactly one of A, B is 1.
2.
Try A = 1, B = 0.
3.
= = 1.
4.
= 0 · 1 = 0.
5.
They differ, so the claim is false. The correct rule is = .

Example

Combining two differences

Is equal to A?

1.
Choose A = 1 and try to make both terms 0: B = 1, C = 1.
2.
= 0 and = 0, so the left side is 0.
3.
The right side is A = 1.
4.
Counterexample found: terms that differ in two variables do not combine.

Common mistakes

Testing only rows where both sides happen to agree, and concluding the claim is true.
Thinking you need several counterexamples. One is a complete disproof.

Practice Counterexample

Interactive questions with instant feedback and a worked solution for every wrong answer.

Boolean Algebra lesson

Which input is a counterexample showing that is not equal to ?

Learn it step by step

Counterexample is taught in Boolean Algebra.

Boolean Algebra →

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