Postulates and Theorems of Boolean Algebra

Boolean algebra is a system of mathematical logic, introduced by a mathematician George Boole in 1854. Boolean algebra differs from ordinary algebra and binary number system. It is a binary algebra defined to perform binary and logical operations.

Take a look at some of the terminologies used in Boolean Algebra, different postulates and theorems of Boolean algebra, which forms the basics for designing a combinational circuits.

Terminologies used in boolean Algebra

Variable – The symbol which represent an arbitrary elements of an Boolean algebra is known as Boolean variable. In an expression, Y=A+BC, the variables are A, B, C, which can value either 0 or 1.

Constant – It is a fixed value. In an expression, Y=A+1, A represents a variable and 1 is a fixed value, which is termed as a constant.

Literal – Each occurrence of a variable in Boolean function either in complemented or normal form is said to be literal.

Postulates of Boolean Algebra

S.No.	Name of the Postulates	Postulate Equation
1	Law of Identity	A + 0 = 0 + A = A A . 1 = 1 . A = A
2	Commutative Law	(A + B) = (B + A) (A . B) = (B . A)
3	Distributive Law	A . (B + C) = (A . B) + (A . C) A + (B . C) = (A + B) . (A + C)
4	Associative Law	A + (B + C) = (A + B) + C (A . B) . C = A . (B . C)
5	Complement Law	A + A’ = 1 A . A’ = 0

Theorems of Boolean Algebra

S.No	Theorem	Statement	Equations
1	Duality Theorem	A boolean relation can be derived from another boolean relation by changing OR sign to AND sign and vice versa and complementing the 0s and 1s.	A + A’ = 1 and A . A’ = 0 are the dual relations.
2	DeMorgan’s Theorem 1	Complement of a product is equal to the sum of its complement.	(A . B)’ = A’ + B’
3	DeMorgan’s Theorem 2	Complement of a sum is equal to the product of the complement.	(A + B)’ = A’ . B’
4	Idempotency Theorem	–	A + A = A A . A = A
5	Involution Theorem	–	A” = A
6	Absorption Theorem	–	A + (A . B) = A A . (A + B) = A
7	Associative Theorem	–	A + (B + C) = (A + B) + C A . (B . C) = (A . B) . C
8	Consensus Theorem	–	AB + A’C + BC = AB + A’C (A + B) + (A’ + C) + (B + C) = (A + B) + (A’ + C)
9	Uniting Theorem	–	AB + AB’ = A (A+B) + (A + B’) = A
10	Other theorems	–	A + 1 = 1 A . 0 = 0
11	Other theorems	–	A + (A’ . B) = A + B A . (A’ + B) = A . B

Solved Examples

Let us solve some examples of boolean function by applying the postulates and theorems of boolean algebra.

Simplify A . (AB + C)

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Simplify A + A’B

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Simplify Y = AB’D + AB’D’

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Simplify Y = (AB’ (C+BD) + A’B’)C