Algebra Know-It-ALL

large, heavy rectangle represents the set of all things that can exist, whether real or imaginary (and
that includes all possible sets). This “emperor of sets” is called the universal set or the universe.
In Fig. 2-1, three of the sets shown inside the universe are finite and two are infinite. Note
how the objects overlap or are contained within one another or are entirely separate. This is
important, because it describes the various ways sets can relate to each other. You can see how
this works by examining the diagram carefully.
All the women in Chicago are people in Illinois, but there are plenty of people in Illinois
who aren’t women in Chicago. The numbers 2, 4, and 6 are positive whole numbers, but there
are lots of positive whole numbers different from 2, 4, or 6. The sets of positive and negative
whole numbers are entirely separate, even though both sets are infinite. None of the positive
or negative whole numbers is a person in Illinois, and no person in Illinois is number (except
according to the government, maybe).

Subsets

When all the elements of a set are also contained in a second set, the first set is called a subset
of the second. If you have two sets A and B, and every element of A is also an element of B,
thenA is a subset of B. That fact can be written

A ⊆ B

How Sets Relate 23

All the

people in

Illinois All the

negative

whole

numbers

All the

women in

Chicago

Universe

All the positive

whole numbers

2

4

6

1/8

1/10

1/12

Figure 2-1 A Venn diagram showing the set of all sets (the

universe) along with a few specific sets within it.