Algebra Know-It-ALL

Listing the elements

When the elements of a set are listed, the list is enclosed in “curly brackets,” usually called

braces. The order of the list does not matter. Repetition doesn’t matter either. The following

sets are all the same:

{1, 2, 3}

{3, 2, 1}

{1, 3, 3, 2, 1}

{1, 2, 3, 1, 2, 3, 1, 2, 3, ...}

The ellipsis (string of three dots) means that the list goes on forever in the pattern shown. In

this case, it’s around and around in an endless cycle.

Now look at this example of a set with five elements:

S= {2, 4, 6, 8, 10}

Are the elements of this set S meant to be numbers or numerals? That depends on the context.

Usually, when you see a set with numerals in it like this, the author means to define the set

containing the numbers that those numerals represent.

Here’s another example of a set with five elements:

P= {Mercury, Venus, Earth, Mars, Jupiter}

You’re entitled to assume that the elements of this set are the first five planets in our solar

system, not the words representing them.

The empty set

A set can exist even if there are no elements in it. This is called the empty set or the null set. It

can be symbolized by writing two braces facing each other with a space between, like this:

{ }

Another way to write it is to draw a circle and run a forward slash through it, like this:

∅

Let’s use the circle-slash symbol in the rest of this chapter, and anywhere else in this book the

null set happens to come up.

You might ask, “How can a set have no elements? That would be like a club with no mem-

bers!” Well, so be it, then! If all the members of the Pingoville Ping-Pong Club quit today and

no new members join, the club still exists if it has a charter and by laws. The set of members

of the Pingoville Ping-Pong Club might be empty, but it’s a legitimate set as long as someone

says the club exists.

Finite or infinite?

Sets can be categorized as either finite or infinite. When a set is finite, you can name all of its

elements if you have enough time. This includes the null set. You can say “This set has no

20 The Language of Sets