Bridge to Abstract Mathematics: Mathematical Proof and Structures

16 SETS Chapter 1

Just as there is an "algebra of numbers" based on operations such as
addition and multiplication, there is also an algebra of sets based on several
fundamental operations of set theory. We develop properties of set algebra
later in this chapter; for now our goal is to introduce the operations by which
we are able to combine sets to get another set, just as in arithmetic we add
or multiply numbers to get a number.

UNION AND INTERSECTION

In the following definitions we assume that all sets mentioned are subsets
of a universal set U.

DEFINITION 1

Let A and B be sets. We define a set formed from A and B, called the intersection

of A and B, denoted A n B (read "A intersection 8") by the rule A n B = {XI x E A

and XE B).

Note that A n B is a set whose elements are the objects common to A

and B; it may be thought of as the "overlap" of A and B.

DEFINITION 2

Let A and B be sets. We define the union of A and 8, denoted A u B (read "A

union B") by A v B = [xjx E A or x E 6).

Again, A u B is a set and is formed from A and B. Its elements are any

objects in either A or B, including any object that happens to lie in both

A and B. (We will see in Chapter 2 that, in mathematical usage, the word

"or" automatically includes the case "or both.") The operations of union

and intersection are called binary operations because they are applied to

two sets to make a third set.

EXAMPLE 1 Let A= {1,3,5,7,9), B= {1,4,7, 10,13,16), and C=

(-5, -3, -1, 1,3,5). Calculate An B, AuB, An C, Bn C, and

B u (A n C).

Solution A n B = (1, 7) since these two objects are common to both sets

and are the only such objects. A u B = (1, 3,4, 5,7,9, 10, 13, 16) since

this set results from "gathering into one set" the elements of A and B.

Similarly, A n C = (1, 3, 5) and B n C = (1 ). To calculate B u (A n C),

we use our result for A n C to arrive at {1,4, 7, 10, 13, 16) u (1, 3, 51,

which equals (1, 3,4, 5, 7, 10, 13, 16).

Note that, in listing the elements of A u B, we write 1 and 7 only once

each, although each occurs in both A and B. As stated earlier, we never list

an object more than once as an element of a set. Also, even though our