Rules of Boolean Algebra

A number of rules of Boolean algebra are available which may be used to simplify Boolean expressions by algebraic simplification.

  1. Basic Concepts
  2. Double Negation
  3. Commutative Law
  4. Associative Law
  5. Distributive Law
  6. Absorption Laws
  7. Idempotence Law
  8. Identity Law
  9. Annulment Law
  10. Complemented Variables
  11. De Morgan’s Theorem

These rules may be confirmed from first principles, or by comparing the equivalent truth tables of the left and right sides of the expression.

The Karnaugh map is an alternative circuit simulation technique, typically used with Boolean expressions having up to 4 input variables.

a: Basic Concepts

A Boolean variable is allowed to have two possible values, which are 0 (false) and 1 (true).

The following basic Boolean relationships may be identified simply by examining the truth tables of the common logic functions AND, OR and NOT.

Basic concepts of Boolean algebra.

b: Double Negation

A double negation cancels out, leaving the original variable.

Double inversion cancels out.
Double negation is equivalent to a pair of inverters in series, which may be removed without altering the logic function of the circuit. This scenario often occurs during NAND-only and NOR-only circuit simplification.

c: Commutative Law

The order of variables is not important for AND or OR operations.

Commutative law.

d: Associative Law

The grouping of terms does not matter where operators have the same level of priority.

Associative law.

e: Distributive Law

The distributive law is analogous to multiplying-out brackets in traditional algebra.

Distributive law.

f: Absorption Laws

The expression is simplified by ‘absorbing’ similar terms.

Absorption law.

g: Idempotence Law

A variable is unchanged when it is ANDed or ORed with itself.

Idempotence law.

h: Identity Law

A variable is unchanged if it is either ANDed with 1 or ORed with 0.

Identity law.

i: Annulment Law

A variable ANDed with 0 gives 0, while a variable ORed with 1 gives 1.

Annulment law.

j: Complemented Variables

ORing a variable with its complement gives 1, while ANDing with a complemented variable gives 0.

Complemented variables.

k: De Morgan’s Theorem

The operation of an AND or OR logic circuit is unchanged if:-
  1. all inputs are inverted,
  2. the operator is changed from AND to OR, or vice versa and,
  3. the output is inverted.

The equivalent Boolean expressions are shown below:-

De Morgan's theorem.

A number of ‘intermediate’ forms may also be encountered, as shown by the following examples

Intermediate forms of De Morgan's theorem.

Worked Example 1

Prove that:-

Worked Example 1. Show / hide answer
Worked Example 1 solution.


Worked Example 2

Prove that:-

Worked Example 2. Show / hide answer
Worked Example 2 solution.

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