Boolean Logic - Boolean Algebra Laws

De Morgan's Law

De Morgan's Law is used to remove negators from brackets. To do this the operator is switched (AND <-> OR) and each argument is negated.

• (a . b)' |=| a' + b'
• (a + b)' |=| a' . b'

Absorbtion

Remove redundant clauses from an expression.

• (a . b) + (a . b . c) |=| a . b
• (a + b) . (a + b + c) |=| a + b
• a + (a . b) |=| a
• a . (a + b) |=| a

Associativity

Removing brackets that have no significance.

• a . (b . c) |=| a . b . c
• a + (b + c) |=| a + b + c

Distributivity

Sometimes described as 'multiplying out the brackets'.

• a . (b + c) |=| (a . b) + (a + c)
• a + (b . c) |=| (a + b) . (a + c)

Idempotency

Idempotency is used to remove repetitions of literals and resolve tautologies and contradictions. A tautology is an equation that always evaluates to true, irrelevant of the values of its arguments (see bullet point 3 below). A contradiction is almost the opposite of a tautology - in that it is an equation that always evaluates to false, irrelevant of the values of its arguments (see bullet point 4 below).

• a . a |=| a
• a + a |=| a
• a + a' |=| true
• a . a' |=| false

Tautologies on Wikipedia
Contradictions on Wikipedia

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