Boolean algebra, a logic algebra, allows the rules used in the algebra of numbers to be applied to logic.
It formalizes the rules of logic. Boolean algebra is used to simplify Boolean expressions
which represent combinational logic circuits.
It reduces the original expression to an equivalent expression that has fewer terms which means that less logic gates are needed to implement
the combinational logic circuit.
Boolean Expression Calculator
Use the calculator to find the reduced boolean expression or to check your (intermediate) answers.
Notes:
Use ~ * + to represent NOT AND OR respectively. Do not omit the * operator for an AND operation.
(~AB)+(B~C)+(AB) will return an error
(~A*B)+(B*~C)+(A*B) is OK
Boolean operations follows a precedence order of NOT AND OR. Expressions inside brackets () are always evaluated ﬁrst, overriding the precedence order.
Please enter variables only, constants like 0,1 are not allowed.
Variables E, I, N, O, Q, S are not allowed
Boolean Expression Simplification
The following example shows how to use algebraic techniques to simplify a boolean expression
~(A * B) * (~A + B) * (~B + B)
~(A * B) * (~A + B) * 1
6 - Complement law
~(A * B) * (~A + B)
5 - Identity law
(~A + ~B) * (~A + B)
8 - DeMorgan's law
~A + ~B * B
4 - Distributive law
~A + 0
6 - Complement law
~A
5 - Identity law
Each line (or step) gives a new expression and the rule or rules used to derive it from the previous one. There can be several ways to arrive at the final result. You can use our calculator to check the intermediate steps of your answer. Equivalent means your answer and the original boolean expression have the same truth table.
Laws of Boolean Algebra
Boolean Algebra Laws are used to simplify boolean expressions.
Basic Boolean Laws
Idempotent Law
A * A = A
A + A = A
Associative Law
(A * B) * C = A * (B * C)
(A + B) + C = A + (B + C)
Commutative Law
A * B = B * A
A + B = B + A
Distributive Law
A * (B + C) = A * B + A * C
A + (B * C) = (A + B) * (A + C)
Identity Law
A * 0 = 0 A * 1 = A
A + 1 = 1 A + 0 = A
Complement Law
A * ~A = 0
A + ~A = 1
Involution Law
~(~A) = A
DeMorgan's Law
~(A * B) = ~A + ~B
~(A + B) = ~A * ~B
Redundancy Laws
Absorption
A + (A * B) = A
A * (A + B) = A
(A * B) + (A * ~B) = A
(A + B) * (A + ~B) = A
A + (~A * B) = A + B
A * (~A + B) = A * B
Each law is described by two parts that are duals of each other. The Principle of duality is
Interchanging the + (OR) and * (AND) operations of the expression.
Interchanging the 0 and 1 elements of the expression.