Solution: In assume that B is a Boolean algebra with

Chapter 6, Problem 3E

(choose chapter or problem)

Problem 3E

In assume that B is a Boolean algebra with operations + and ∙. Give the reasons needed to fill in the blanks in the proofs, but do not use any parts of Theorem unless they have already been proved. You may use any part of the definition of a Boolean algebra and the results of previous exercises, however.

Theorem Double Complement Law

For all elements a in a Boolean algebra B,  = a.

Proof:

Suppose B is a Boolean algebra and a is any element of B. Then

and

Thus a satisfies the two equations with respect to  that are satisfied by the complement of . From the fact that the complement of a is unique, we conclude that  = a.

Exercise

For all a and b in B, (a + b) ∙ a = a.

Proof: Let a and b be any elements of B. Then

Exercise

In assume that B is a Boolean algebra with operations + and ∙. Give the reasons needed to fill in the blanks in the proofs, but do not use any parts of Theorem 6.4.1 unless they have already been proved. You may use any part of the definition of a Boolean algebra and the results of previous exercises, however.

Theorem Double Complement Law

For all elements a in a Boolean algebra B,  = a.

Proof:

Suppose B is a Boolean algebra and a is any element of B. Then

and

Thus a satisfies the two equations with respect to  that are satisfied by the complement of . From the fact that the complement of a is unique, we conclude that  = a.

Exercise

For all a in B, a + 1 = 1.

Proof: Let a be any element of B. Then

Example

Proof of an Idempotent Law

Fill in the blanks in the following proof that for all elements a in a Boolean algebra B, a + a = a.

Proof:

Suppose B is a Boolean algebra and a is any element of B. Then

Solution

a. because 0 is an identity for +

b. by the complement law for ∙

c. by the distributive law for + over ∙

d. by the complement law for +

e. because 1 is an identity for ∙