Boolean Algebra Complement

Boolean algebra operators are AND, OR, NOT and equal. The process of NOT operator is found the complement or opposite. Through the truth table we can learn Boolean algebra operation easily. Truth table is connected with the Boolean algebra. In the truth table the expression ‘true is represented as ‘T and the expression ‘false is represented as ‘F. Let us see Boolean algebra complement and their symbols in this article.
Boolean Algebra Complement:

Only one operand is present in the NOT operator. For example ~X here the symbol ~ is used to denote expression opposite or complement. The most important operation of operator NOT is complement or opposite.

Some rules in Boolean algebra complement:

Complement of PQ:

(PQ) = P + Q

Complement of P Q

(P Q) =P + Q

Complement of P + Q + R

(P + Q + R) = P * Q * R

Complement of PQR:

(PQR) = P + Q + R

Easy way to remember this Complement rule is put * for + and put + for *

For example:

Complement of PQ = > (PQ) = P + Q
Proof for Boolean Algebra Complement:

Prove:

Complement of P Q `!=` PQ

Proof:

Given

(P Q) `!=` PQ

LHS:

P Q

Let us consider P=1 and Q =0

F = P Q

F = 1 0

F= 0 1

F = 0

Take the complement of F

We get

F =1

RHS:

PQ

P=1 and Q =0

F = 1 0

F =0

LHS is not equal to RHS therefore (P Q) `!=` PQ

Prove:

Complement of P Q = P + Q

Proof:

Given

(P Q) = P + Q

LHS:

P Q

Let us consider P=1 and Q =0

F = P Q

F = 1 0

F= 0 1

F =0

Take the complement of F

We get

F =1

RHS:

P+Q

P=1 and Q =0

F = 1 + 0

F =1

LHS is equal to RHS therefore (P Q) = P + Q.

Prove:

(PQ) = P + Q

Proof:

Given:

(PQ) = P + Q

LHS:

(PQ)

Let us consider P=1 and Q =0

F = (1 0)

F = (0)

F =1

RHS:

P + Q

Let us consider P=1 and Q =0

F =1 + 0

F =0 + 1

F = 1

LHS is equal to RHS therefore (PQ) = P + Q