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Set Notation

You never know when set notation is going to pop up. Usually, you'll see it when you learn about solving inequalities, because for some reason saying "x < 3" isn't good enough, so instead they'll want you to phrase the answer as "the solution set is { x | x is a real number and x < 3 }". How this adds anything to the student's understanding, I don't know. But I digress....

A set, informally, is a collection of things. The "things" in the set are called the "elements", and are listed inside curly braces. For instance, if I were to list the elements of "the set of things on my kid's bed when I wrote this lesson", the set would look like this:

{pillow, rumpled bedspread, a stuffed animal, one very fat cat who's taking a nap}

Sets are usually named using capital letters. This isn't a rule, as far as I know, but it does seem to be traditional. So let's name this set as "A". Then we have:

A = {pillow, rumpled bedspread, a stuffed animal, one very fat cat who's taking a nap}

The cat's name is "Junior", so this set could also be written as:

A = {pillow, rumpled bedspread, a stuffed animal, Junior}

Sets are "unordered", which means that the things in the set do not have to be listed in any particular order. The set above could just as easily be written as:

A = {Junior, pillow, rumpled bedspread, a stuffed animal}

We use a special character to say that something is an element of a set. It looks like an odd curvy capital E. For instance, to say that "pillow is an element of the set A", we would write the following:

This is pronounced "pillow is an element of A".

The elements of a set can be listed out according to a rule, such as:

{x is a natural number, x < 10}

If you're going to be technical, you can use full "set-builder notation", which looks like this:

This is pronounced as "the set of all x, such that x is an element of the natural numbers and x is less than 10". The vertical bar is usually pronounced as "such that", and it comes between the name of the variable you're using to stand for the elements and the rule that tells you what those elements actually are. This same set, since the elements are few, can also be given by a listing of the elements, like this:

{1, 2, 3, 4, 5, 6, 7, 8, 9}

Listing the elements explicitly like this, instead of using a rule, is often called "using the roster method".

Your text may or may not get technical regarding the names of the types of numbers. If it does, these are the symbols to use:

Yes, the symbols require those double-barred strokes.

Sets can be related to each other. If one set is "inside" another set, it is called a "subset". Suppose A = {1, 2, 3} and B = {1, 2, 3, 4, 5, 6}. Then A is a subset of B, since everything in A is also in B. This is written as:

That sideways-U thing is the subset symbol, and is pronounced "is a subset of". To show something is not a subset, you draw a slash through the subset symbol, so the following:

...is pronounced as "B is not a subset of A".

If two sets are being combined, this is called the "union" of the sets, and is indicated by a large U-type character. If instead of taking everything from the two sets, you're only taking what is common to the two, this is called the "intersection" of the sets, and is indicated with an upside-down U-type character. So if C = {1, 2, 3, 4, 5, 6} and D = {4, 5, 6, 7, 8, 9}, then:

These are pronounced as "C union D equals..." and "C intersect D equals...", respectively.

• Give a solution using the roster method:
  A = { 1, 2, 3, 4, 5, 6, 7 }, B is a subset of A, the elements of B are even.

The numbers in A that are even are 2, 4, and 6, so B = {2, 4, 6}.

• What is the intersection of A = { x is odd } and B = { x is between –4 and 6 }, where the elements of the two sets are integers?

Since "intersection" means "only things that are in both sets", the intersection will be all the numbers which are both odd and between –4 and 6.

{–3, –1, 1, 3, 5}

• What is the union of A = { x is a natural number between 4 and 8 inclusive }
  and B = { x is a single-digit negative integer }?

Since "union" means "anything that is in either set", the union will be everything from A plus everything in B. Since A = { 4, 5, 6, 7, 8 } and B = { –9, –8, –7, –6, –5, –4, –3, –2, –1 }, then their union is:

{ –9, –8, –7, –6, –5, –4, –3, –2, –1, 4, 5, 6, 7, 8 }

• Give a solution using a rule: The set of all the odd integers.

An odd integer is one more than an even integer, and every even integer is a multiple of 2, so the set would be: Copyright © Elizabeth Stapel 2005-2011 All Rights Reserved

This is pronounced as "all integers x such that x is equal to 2 times m plus 1, where m is an integer", which is a fancy way of saying "x is an odd integer". It's a lot easier to describe this set using the roster method: {..., –3, –1, 1, 3, 5, 7, ...}. The ellipsis (the three periods in a row) means "and so forth", and indicates that the pattern continues indefinitely in the given direction.

There's plenty more you can do with set notation, but the above is usually enough to get by in most algebra-class circumstances. If you need more, try doing a web search for "set notation".

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