Algebra Know-It-ALL

Figure 2-1 shows that the set of all the women in Chicago is a subset of the set of all the

people in Illinois. That is expressed by a hatched square inside a shaded oval. Figure 2-1 also

shows that the set {2, 4, 6} is a subset of the set of positive whole numbers. That is expressed

by placing the numerals 2, 4, and 6 inside the rectangle representing the positive whole num-

bers. All five of the figures inside the large, heavy rectangle of Fig. 2-1 represent subsets of the

universe. Any set you can imagine, no matter how large, small, or strange it might be, and no

matter if it is finite or infinite, is a subset of the universe. Technically, a set is always a subset

of itself.

Often, a subset represents only part, not all, of the main set. Then the smaller set is called

aproper subset of the larger one. In the situation shown by Fig. 2-1, the set of all the women in

Chicago is a proper subset of the set of all the people in Illinois. The set {2, 4, 6} is a proper sub-

set of the set of positive whole numbers. All five of the sets inside the main rectangle are proper

subsets of the universe. When a certain set C is a proper subset of another set D, we write

C⊂D

Congruent sets

Once in a while, you’ll come across two sets that are expressed in different ways, but they turn

out to be exactly the same when you look at them closely. Consider these two sets:

E= {1, 2, 3, 4, 5, ...}

F= {7/7, 14/7, 21/7, 28/7, 35/7, ...}

At first glance, these two sets look completely different. But if you think of their elements as

numbers (not as symbols representing numbers) the way a mathematician would regard them,

you can see that they’re really the same set. You know this because

7/7 = 1

14/7= 2

21/7= 3

28/7= 4

35/7= 5

↓

and so on, forever

Every element in set E has exactly one “mate” in set F, and every element in set F has exactly

one “mate” in set E. In a situation like this, the elements of the two sets exist in a one-to-one

correspondence.

When two sets have elements that are identical, and all the elements in one set can

be paired off one-to-one with all the elements in the other, they are said to be congruent

sets. Sometimes they’re called equal sets or coincident sets. In the above situation, we can

write

E=F

24 The Language of Sets