Chapter 6.4, Problem 7ES

Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

Chapter
Section

Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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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 determine

To Prove:

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

Explanation

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=aa¯     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 aB.

a1=a and a1=a

It follows that,

a+1=a+1

Add both sides with a¯ ,

a¯+(a1)=a¯+(

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