Set Notation
Purplemath
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 was "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 as "pillow is an element of A".
The elements of a set can be listed out according to a rule, such as:
{ x is something on my kid's bed }
A mathematical example of a set whose elements are named according to a rule might be:
{x is a natural number, x < 10}
If you're going to be technical, you can use full "set-builder notation" to express the above mathematical set. In set-builder notation, the previous set looks like this:
The above 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:
: the natural numbers
: the integers
: the rationals
: the real numbers
Yes, the symbols require those double-barred strokes for all the vertical portions of the characters.
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 relationship 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:
{ 1, 2, 3, 4, 5, 6, 7, 8, 9 }
{ 4, 5, 6 }
These are pronounced as "C union D equals..." and "C intersect D equals...", respectively.
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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 set B is a subset of A, so it contains only things that are in A. The elements of B are even, so I need to pick out the elements of A which are even; these will be the elements of the subset B.
The numbers in A that are even are 2, 4, and 6, so:
B = { 2, 4, 6 }.
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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 in each of the sets. The elements of B can be listed, being not too many integers:
B = { −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, 6 }
The elements of A are all the odd integers. There are infinitely-many of them, so I won't bother with a listing. The intersection will be the set of integers which are both odd and also between −4 and 6. In other words:
{ −3, −1, 1, 3, 5 }
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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 } (because "inclusive" means "including the endpoints") 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 }
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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. The formal way of writing "is a multiple of 2" is to say that something is equal to two times some other integer; in other words, "x = 2m", where "m" is some integer. Then an odd integer, being one more than a multiple of 2, is x = 2m + 1.
So, in full formality, the set would be written as:
The solution to the example above is pronounced as "all integers x such that x is equal to 2 times m plus 1, where m is an integer".
It's a lot easier to describe the last set above using the roster method:
{ ..., −3, −1, 1, 3, 5, 7, ... }
The ellipsis (that is, the three periods in a row) means "and so forth", and indicates that the pattern continues indefinitely in the given direction. Or, if the dots are between elements, like this:
{ 0, 3, 6, 9, ..., 993, 996, 999 }
...it means that the pattern continues in the same manner through the unwritten middle.
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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