In 4—10 assume that B is a Boolean algebra with operations + and•. Prove each statement using only the axioms for a Boolean algebra and statements proved in the text or in lower-numbered exercises.

5. Complements of 0 and 1:

a. 0 ¯ = 1

b. 1 ¯ = 0

(a)

To Prove:

0¯=1

Given information:

Let B be the Boolean algebra, with the operations, addition "+" and multiplication ".".

Concept used:

a¯+a=a+a¯     by the commutative law=1            by the complement law for 1

And

a¯⋅a=a⋅a¯     by the commutative law=0            by the complement law for 0

Calculation:

Let B be the Boolean algebra, with the operations, addition "+" and multiplication "."

The identity element with respect to addition is 0 and the identity element with respect to multiplication is 1.

The uniqueness or the complement law tells that, for all a and x in B, if a+x=1 and a

(b)

To Prove:

1¯=0