Bridge to Abstract Mathematics: Mathematical Proof and Structures

12 SETS Chapter 1

Since "element hood" and "subset" are both relationships of containment,

it is important not to confuse the two. For instance, the set A = {I, 2,3)

contains {2,3) and contains 2, but in different senses. As another example,

we note that 3 E A and (3) G A are true statements, whereas (3) E A and

3 c A are false. Again, a basic feature of good mathematical writing is

precision.

There is a special danger of confusion in dealing with the subset relation-

ship in connection with the empty set @. Consider the question whether

@ E A and/or @ c A, where A = {1,2,3). These questions, especially

the second, in view of the criterion given in Remark 3, are not easy to decide.

This difficulty is one of several we meet in this chapter that highlight the need

for a background in logic and proof. Chapters 2 through 4, providing such

a background, discuss a more formal and precise approach to some of the

informal "definitions" in this chapter, including that of Remark 3.

Finally, in view of our alternative statement (1) of the criterion for set

equality in Remark 2, the connection between the relationships "subset"

and "equality of sets" is: A = B if and only if A r B B c A. This is a

crucial fact that We will prove rigorously in Chapter 4 (Example 3, Article

4.1) and use repeatedly in formulating proofs from that point through the

remainder of the text.

Proper subset. When we are told that A G B, the possibility that A and B

are equal is left open. To exclude that possibility, we use the notation and

terminology of proper subset.

DEFINITION 4

Let A and 6 be sets. We say that A is a proper subset of 8, denoted A c B, if

and only if A c B, but A # B. We write A $ B to symbolize the statement that

A is not a proper subset of B (which could mean that either A $ B or A = 6).

EXAMPLE 4 Explore various subset and proper subset relationships among
the sets A = (1,2, 31, B = (l,2, 3,4), C = (2, 3, I), and D = (2,4, 6).
7-

Solution The subset relationships are A c B, C G B, A c C, and C c A.

As for proper subset yelationships, we have A c B and C c B. Note,

however, that A is not a proper subset of C (nor C of A) since A and C

are equal. Finally, note that even though A and D are not equal, A is not

a proper subset of D since A is not a subset of D.

Some texts use the notation A c B to denote "subset," at the same time

using A B to denote "proper subset." This is an example of a problem

with which all mathematicians and students of mathematics must deal,

namely, the widespread nonuniformity of mathematical notation. The best

rule to remember is that the burden of correct interpretation rests on you!