Boolean Algebra: Theorem

The theorems used to change the form of boolean algebra expression are known as Boolean algebra theorems. They are as follows:

De Morgan’s Theorem

According to De Morgan’s theorem, 

1. (A . B)’ = A’ + B’ 
2. (A + B)’ = A’ . B’ 

Transposition Theorem

According to the Transposition Theorem,

AB + A’C = (A + C) (A’ + B)

The proof for the above mentioned theorem is as follows:

RHS 

= (A + C) (A’ + B)

= AA’ + A’C + AB + CB

= 0 + A’C + AB + BC

= A’C + AB + BC(A + A’)

= AB + ABC + A’C + A’BC

= AB + A’C

= LHS

Complementary Theorem

One can obtain the complementary expression by doing the following:

1. Change OR sign by AND sign and vice versa. 
2. Any 0 and 1 expression must be complemented. 
3. The individual literals is complemented. 

For example:

Complement of A(B+C) = A’+(B’.C’) = (A’+B’)(A’+C’) 

Redundancy theorem

Redundancy theorem is used to eliminate redundant terms. 

For example:

AB + BC’ + AC = AC + BC’

Proof:

LHS 

= AB + BC’ + AC

= AB(C + C’) + BC'(A + A’) + AC(B + B’)

= ABC + ABC’ + ABC’ + A’BC’ + ABC + AB’c

= ABC + ABC’ + A’BC’ + AB’C

= AC(B + B’) + BC'(A + A’)

= AC + BC’

= RHS

Duality theorem

Duality is equivalent to writing a negative logic for a boolean expression. 

Reference

Boolean Algebra: Theorem