Section 7.1: Boolean Algebra Structure

Abstract:

First of all, note that we're only reading 7.1 through p. 468/541 (up to Isomorphic Boolean Algebras).

A Boolean algebra (named after George Boole ) is a generalization and an abstraction of the propositional logic we studied early this term, as well as the set theory which we glanced at, albeit briefly. We are really interested in using it to understand the basic elements of computer logic, however, which is based on a binary (0,1) alphabet. In this first section we are merely introduced to the fundamental concepts of Boolean Algebra.

Definition and Terminology

Definition: a Boolean Algebra is a set B on which are defined two binary operations + and , and one unary operation ', and in which there are two distinct elements 0 and 1 such that the following properties hold for all :

The element x' is called the complement of x. The algebra may be denoted .

Of these properties, certainly the distributive property 3a. may seem the strangest, since it obviously doesn't hold for the usual suspects + and .

Notice the beautiful symmetry in this definition: the roles of + and are exactly reversed with respect to the special elements 0 and 1.

Question: how are these reflected in the properties of propositional logic that we studied earlier this term?

• A comment on functions , and example (implication).
• The difference between and =....

In Example 2, p. 465/538, the set consisting of only two elements (so they must be our distinguished elements), and the binary operations of + by x+y=max(x,y) and that of by . Complements are given by 0'=1 and 1'=0. It turns out that this is another example of a Boolean Algebra.

Example: Practice 1, p. 465/538

Curiously enough, x+x=x in a Boolean Algebra (this is the idempotent property. You'll want to remember that one, for any proofs!) And since x+x=x, we must have by the beautiful symmetry of the operations. This symmetry, known as duality, means that we only have to do half the work most of the time....

You may have bumped into this concept in linear algebra: for example, projection matrices are idempotent, such as

This matrix projects onto the first, third, and fourth dimensions; and projecting onto those dimensions a second time doesn't change anything (i.e., ).

Example: Practice 2, p. 467/540

Given an element x of the set B of a Boolean Algebra, the complement x' is the unique element of B with the property that

Example: Practice 3, p. 467/540

Hints for proving Boolean Algebra Equalities (p. 467/540):

• Usually the best approach is to start with the more complicated expression and try to show that it reduces to the simpler expression.
• Think of adding some form of 0 (like ) or multiplying by some form of 1 (like x+x').
• Don't forget property 3a, the distributive property of addition over multiplication, just because it seems weird!
• Remember the idempotent properties: x+x=x and .
• It may help to frame the argument in terms of either set theory or propositional logic, to give you a framework for understanding the thrust.

Example: Exercise 8/9, p. 475/548

Example: Exercise 11a/12a, p. 476/548