Simplifying Boolean expressions using Boolean identities and rules

Understanding Boolean Identities And Rules

Boolean identities and rules are fundamental concepts in logic, providing the basis for simplifying Boolean expressions.
They can be likened to algebraic laws (like the commutative law in maths) but apply specifically to Boolean algebra.

Common Boolean Identities

There are several standard identities in Boolean algebra. Among the most common are Identity Law, Null Law, Idempotent Law, Complementation Law, Double Negation Law, Commutative Law, Associative Law, Distributive Law, Absorption Law and De Morgan’s Law.

Identity Law

The Identity Law states that any value ANDed with true will be the value itself (A AND true = A), and any value ORed with false will be the value itself (A OR false = A).

Null Law

The Null Law holds that any value ANDed with false will result in false (A AND false = false) and any value ORed with true will equate to true (A OR true = true).

Idempotent Law

As per the Idempotent Law, any value ANDed or ORed with itself will result in the value itself (A AND A = A and A OR A = A).

Complementation Law

The Complementation Law posits that ORing a value with its negation yields true, while ANDing a value with its negation produces false (A OR NOT A = true and A AND NOT A = false).

Double Negation Law

The Double Negation Law simply indicates that the NOT of the NOT of a variable is the variable itself (i.e., NOT (NOT A) = A).

Commutative Law, Associative Law and Distributive Law

Commutative Law establishes that the order of operations doesn’t affect the outcome in AND and OR operators. A AND B is the same as B AND A and likewise for OR.
Associative Law suggests that grouping doesn’t change the result. (A AND B) AND C equals A AND (B AND C). The same holds true for OR operations.
The Distributive Law points to the fact that A AND (B OR C) equals (A AND B) OR (A AND C), and similarly, A OR (B AND C) equals (A OR B) AND (A OR C).

Absorption Law and De Morgan’s Law

The Absorption Law indicates that A AND (A OR B) is equal to A, as is A OR (A AND B).
De Morgan’s Law states that NOT (A AND B) is the same as (NOT A OR NOT B), and NOT (A OR B) equals (NOT A AND NOT B).

Simplifying Boolean Expressions

Applying these Boolean identities and laws to Boolean expressions can aid in simplifying complex expressions.
Simplifying boolean expressions can make them easier to understand and more efficient in terms of computational resources.
Simplification often means reducing the number of logic gates in a digital circuit, thus making the circuit simpler, cheaper, and more reliable.
Familiarise yourself with these identities and rules until you can recognise and apply them almost automatically. Practice makes perfect!