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
|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
Let us solve some examples of boolean function by applying the postulates and theorems of boolean algebra.
Simplify A . (AB + C)
Simplify A + A’B
Simplify Y = AB’D + AB’D’
Simplify Y = (AB’ (C+BD) + A’B’)C