Algebra Know-It-ALL

Are you confused?

Do you still wonder what makes a bunch of things a set? If you have a basket full of apples and you call it

a set, is it still a set when you dump the apples onto the ground? Were those same apples elements of a set

before they were picked? Questions like this can drive you crazy if you let them. A collection of things is a

set if you decide to call it a set. It’s that simple.

As you go along in this course, you’ll eventually see how sets are used in algebra. Here’s an easy example.

What number, when multiplied by itself, gives you 4? The obvious answer is 2. But −2 will also work,

because “minus times minus equals plus.” In ordinary mathematics, a number can’t have more than one

value. But two or more numbers can be elements of a set. A mathematician would say that the solution set

to this problem is {−2, 2}.

Here’s a riddle!

You might wonder if a set can be an element of itself. At first, it is tempting to say “No, that’s impossible.

It would be like saying the Pingoville Ping-Pong Club is one of its own members. The elements are the

Ping-Pong players, not the club.”

But wait! What about the set of all abstract ideas? That’s an abstract idea. So a set can be a member of

itself. This is a strange scenario because it doesn’t fit into the “real world.” In a way, it’s just a riddle. Nev-

ertheless, riddles of this sort sometimes open the door to important mathematical discoveries.

Here’s a challenge!

Define the set of all the positive and negative whole numbers in the form of an “implied list” of numerals.

Make up the list so that, if someone picks a positive or negative number, no matter how big or small it

might be, you can easily tell whether or not it is in the set by looking at the list.

Solution

You can do this in at least two ways. You can start with zero and then list the numerals for the positive and

negative whole numbers alternately:

{0, 1, −1, 2, −2, 3, −3, 4, −4, ...}

You can also make the list open at both ends, implying unlimited “travel” to the left as well as to the

right:

{...,−4,−3,−2,−1, 0, 1, 2, 3, 4, ...}

How Sets Relate

Now let’s see how sets can be broken down, compared, and combined. Pictures can do the

work of thousands of words here.

Venn diagrams

One of the most useful illustrations for describing relationships among sets is a Venn diagram, in

which sets are shown as groups of points or as geometric figures. Figure 2-1 is an example. The

22 The Language of Sets