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Null law

Also called: domination law, dominance law, annulment law, null element, null laws, null property

The Boolean laws A + 1 = 1 and A · 0 = 0: a 1 in an OR, or a 0 in an AND, fixes the result no matter what A is.

The null law (also called the domination or annulment law) is the pair:

  • = 1
  • = 0

Intuition: OR asks "is anything 1?" If one input is already 1, the answer is yes whatever A is. AND asks "is everything 1?" If one input is 0, the answer is no.

In a circuit, an OR gate with an input stuck at 1 always outputs 1, and an AND gate with an input stuck at 0 always outputs 0. The other inputs no longer matter.

The null law is the heart of the proof of the absorption law: = = = A, where = 1 is the null law.

Compare it with the identity law: 0 does nothing to an OR but dominates an AND, and 1 does nothing to an AND but dominates an OR.

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Worked example

Example

A term that wipes everything out

Simplify .

  1. 1.

    = 0 (null), so the first bracket is = A (identity).

  2. 2.

    = 1 (null).

  3. 3.

    The expression is now = A (identity).

  4. 4.

    Result: A. Neither B nor C matters.

Common mistakes

  • Writing = A. That is the identity law with the wrong constant; ORing with 1 gives 1.

  • Treating + as addition and getting 1 + 1 = 2. In Boolean algebra 1 + 1 = 1.

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Null law is taught in Boolean Algebra.

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