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.

7. Uniqueness of 1: There is only one element of B that is an identity for •.

To Prove:

There is only one element of B that is an identity for.

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 "⋅".

Suppose 1 and 1` are tile identity elements of B with respect to multiplication "⋅".

To show that there is only identity element of B with respect to multiplication "⋅" it is enough to show that 1=1`.

The Identity law for multiplication "⋅" tells that, for all a∈B.

a⋅1=a and a⋅1`=a

It follows that,

a+1=a+1`

Add both sides with a¯ ,

a¯+(a⋅1)=a¯+(