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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
elements,” and you’ve named all the elements of the null set. When a set is infinite, you can’t name all of its elements, no matter how much time you have.
Even if a set is infinite, you might be able to write an “implied list” that reveals exactly what all of its elements are. Consider this:
W= {0, 1, 2, 3, 4, 5, ...}
This is the set of whole numbers as it is usually defined in mathematics. You know whether or not something is an element of set W, even if it is not shown above, and even if you could not reach it if you started to scribble down the list right now and kept at it for days. You can tell right away which of the following numbers are elements of W, and which are not:
12 1/2 23 100/3 78,883,505
356.75
90,120,801,000,000,000
−65,457,333
The first, third, fifth, and seventh numbers are elements of W, but the second, fourth, sixth, and eighth numbers are not.
Some infinite sets cannot be totally defined by means of any list, even an “implied list”!
You’ll learn about this type of set in Chap. 9.
Sets within sets
A set can be an element of another set. Remember again, anything can be a member of a set!
You can have sets that get confusing when listed. Here are some examples, in increasing order of strangeness:
{1, 2, 3, 4, 5}
{1, 2, {3, 4, 5}}
{1, {2, {3 ,4, 5}}}
{1, {2, {3, {4, 5}}}}
{1, {2, {3, {4, {5}}}}}
An “inner” or “member” set can sometimes have more elements than the set to which it belongs. Here is an example:
{1, 2, {3, 4, 5, 6, 7, 8}}
Here, the main set has three elements, one of which is a set with six elements.
The Concept of a Set 21
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, ...}