Bridge to Abstract Mathematics: Mathematical Proof and Structures

20 SETS Chapter 1

EXAMPLE 6 Let A = {1,3,5,7), B = {1,2,4,8,16), and C = {1,2,3,... ,

100). Is 1 an element of A - B? Compute A - B, B - A, B - C, and

A - C.

Solution Since 1 E A, there is a possibility that 1 could be in A - B. But

the fact that 1 is also an element of B rules this out; that is, 1 4 A - B.

In fact, to compute A - B, we simply remove from A any object that

is also in B. Hence A - B = (3, 5, 7). Similarly, B - A = (2,4, 8, 161,

whereas A - C = B - C = @. 0

What general conclusions can be guessed from our calculation of A - B

and B - A in Example 6? Do the results of calculating A - C and B - C

suggest any possible general facts? Describe the sets C - A and C - B?

Can you compute A - B' from the information given?

Using the sets from Example 4, we note that A - B = [-I, -41,

B - A = (1,2]. Calculate also A - C, C - D, C - B, A - @, and 0 - D?

Are you willing to speculate on any further general properties of difference

based on these results?

SYMMETRIC DIFFERENCE

Very often in mathematics, once a certain body of material (e.g., definitions

and/or theorems) has been built up, the work becomes easier. New defini-

tions can be formulated in terms of previous ones, rather than from first

principles, and proofs of theorems are frequently shorter and less laborious

once there are earlier theorems to justify or eliminate steps. The first exam-

ple of this situation occurs now in the definition of our fifth operation on sets,

symmetric diflerence.

DEFINITION 5

Let A and 8 be sets. We define the symmetric difference of A,and 6, de-

noted A A 8, by the rule A A B = (A - B) u (B - A).

Note that we have not defined this operation using set-builder notation.

Rather, we have used a formula that employs previously defined set opera-

tions. This approach has advantages and disadvantages as compared to a

definition from first principles. Advantages include compactness and math-

ematical elegance, which make this type of definition more pleasing to ex-

perienced readers. The major disadvantage, however, affecting primarily

the less experienced, is that this type of definition usually requires some

analysis in order to be understood. In this case we must analyze carefully

the right-hand side of the equation, the dejning rule for the operation.

To be in A A B, an object must lie either in A - B or B - A (or both?

1
Which objects are in both A - B and B - A?), that is, either in A but not