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Home / Boolean Algebra / Laws of Boolean Algebra

Laws of Boolean Algebra
Boolean Algebra uses a set of Laws and Rules to define the operation of a digital logic circuit


As well as the logic symbols “0” and “1” being used to represent a digital input or output, we can also use them as constants for a permanently “Open” or “Closed” circuit or contact respectively.

A set of rules or Laws of Boolean Algebra expressions have been invented to help reduce the number of logic gates needed to perform a particular logic operation resulting in a list of functions or theorems known commonly as the Laws of Boolean Algebra.

Boolean Algebra is the mathematics we use to analyse digital gates and circuits. We can use these “Laws of Boolean” to both reduce and simplify a complex Boolean expression in an attempt to reduce the number of logic gates required. Boolean Algebra is therefore a system of mathematics based on logic that has its own set of rules or laws which are used to define and reduce Boolean expressions.

The variables used in Boolean Algebra only have one of two possible values, a logic “0” and a logic “1” but an expression can have an infinite number of variables all labelled individually to represent inputs to the expression, For example, variables A, B, C etc, giving us a logical expression of A + B = C, but each variable can ONLY be a 0 or a 1.

Examples of these individual laws of Boolean, rules and theorems for Boolean Algebra are given in the following table.

Truth Tables for the Laws of Boolean
Boolean
Expression Description Equivalent
Switching Circuit Boolean Algebra
Law or Rule
A + 1 = 1 A in parallel with
closed = "CLOSED" universal parallel circuit Annulment
A + 0 = A A in parallel with
open = "A" universal parallel Identity
A . 1 = A A in series with
closed = "A" universal series circuit Identity
A . 0 = 0 A in series with
open = "OPEN" universal series Annulment
A + A = A A in parallel with
A = "A" idempotent parallel circuit Idempotent
A . A = A A in series with
A = "A" idempotent series circuit Idempotent
NOT A = A NOT NOT A
(double negative) = "A" Double Negation
A + A = 1 A in parallel with
NOT A = "CLOSED" complement parallel circuit Complement
A . A = 0 A in series with
NOT A = "OPEN" complement series circuit Complement
A+B = B+A A in parallel with B =
B in parallel with A absorption parallel circuit Commutative
A.B = B.A A in series with B =
B in series with A absorption series circuit Commutative
A+B = A.B invert and replace OR with AND de Morgan’s Theorem
A.B = A+B invert and replace AND with OR de Morgan’s Theorem