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Grade 8 · lesson 3 of 4

Check Whether Two Rules Are Equivalent

Use every Boolean input combination to compare two expressions.

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Supplied materials

  • Paper and pencil; an adult may read aloud. No device, account or purchase is needed.

Learn the idea

Two Boolean rules are equivalent when they produce the same result for every allowed input combination. A few matching examples do not establish equivalence. With two Boolean inputs there are four combinations; enumerate all four exactly once. A row where the outputs differ is a counterexample.

Compare rule R = NOT (A OR B) with rule S = (NOT A) AND (NOT B). Make rows (false,false), (false,true), (true,false), (true,true). Compute each rule independently before comparing columns. This tests a small instance of a logical equivalence rather than assuming similar words mean the same rule.

  1. Write all four input pairs without duplicates.
  2. Evaluate A OR B, then reverse it for R.
  3. Reverse A and B separately, then combine with AND for S.
  4. Compare R and S on every row, and contrast NOT (A AND B) as a different rule.
Worked example

See the method in action

R outputs true, false, false, false. S outputs true, false, false, false, so they are equivalent for all four Boolean pairs. NOT (A AND B) outputs true, true, true, false and is different.

Your turn

Check your reasoning

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Which pair shows that NOT (A OR B) differs from NOT (A AND B)?

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See the answer and explanation

A = false, B = true

At false,true, NOT OR is false while NOT AND is true. The all-false and all-true pairs give matching outputs and would miss the difference.

Take it further

Change S to (NOT A) OR (NOT B). Expected column: true, true, true, false. It no longer matches R on the mixed-input rows.

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