Set notation is a precise way of describing which items belong in a group and which do not.A set can contain any sort of item, but here we'll talk about sets of numbers.

Very specific symbols are used in set notation and math professors are notorious for being sticklers for accuracy in the formation of these symbols when you write them, and for placing them in the proper order. However, the concepts of set notation do not require genius level intelligence or a lot of time to master.

A "set" of numbers is simply any group of numbers that you want to define in exclusion of other numbers. For instance the set of integers from 1 to 3 includes the numbers 1, 2 and 3, but excludes any number less than 1 or greater than 3, any numbers between 1 and 2, or between 2 and 3.

But that's a lot of words, and your math professor is not interested in an essay on the subject. So set notation has been devised to say all that much more succinctly. The two main methods of set notation are the roster method of set notation and set builder notation.

The roster method of set notation requires only a few of the symbols you see to the right. Set builder notation requires slightly more. The rest are listed here so that you can see how many different types of sets can be described with set notation.

In both the roster method of set notation and set builder notation, each set is given a name. Sets are usually named with capital letters. We'll call our example set "A".

1. Write an English language sentence that describes the set of numbers you want to notate.

• Example: "A is the set of all integers from one to ten".

2. Using the symbols from the list, translate the English language description of your set into set notation. For the roster method of set notation, as the name suggests, simply list the elements of the set.

• Example: A = {1,2,3,4,5,6,7,8,9,10}

This set notation is read "The set A consists of the numbers one, two, three, four, five, six, seven, eight, nine and ten."

The roster method of set notation can also be used to describe an infinite set of numbers.

• Example: A = {1,2,3,4,5,6,7,8,9,10...}

Note the "..." at the end of the list. This set notation is read "The set A consists of the numbers one, two, three, four, five, six, seven, eight, nine, ten and so forth", meaning that the set elements continue infinitely in the established pattern.

3. Use set builder notation when a set contains too many numbers to list conveniently. When using set builder notation, give a generic name to the elements of the set. Usually a lower case letter is used; x is a perennial favorite, but other letters are also used.

In set builder notation we use fancy math jargon like "all real numbers x such that..." which is denoted with "x|". Also, frequently you will need to specify that your set elements belong to another previously defined set such as the set of integers, like our example. This is denoted as "x ε ℤ", which reads, "x is an integer" or "all integers x". The same example set A that is described before with the roster method of set notation is described in set builder notation as follows:

• Example: A = {x ε ℤ | 0 ≤ x ≥ 10}

This set notation is read "The set A consists of all integers x such that zero is less than or equal to x and x is greater than or equal to ten."

4. To demonstrate how set builder notation is useful for describing large or infinite sets of numbers, here are is another example:

• Example: B = {y ε ℤ | 0 ≤ y }

This set notation is read "The set B consists of all integers y such that zero is less than or equal to y." This set contains an infinite number of elements-all integers greater than zero.

5. To express relationships between sets, use these symbols as shown in the following examples:

• ∪ the union of
• ∩ the intersection of
• ⊂ is a subset of
• ⊄ is not a subset of

• Example: ℕ ∩ {0,½,1,3,4,5}= {1,3,4,5}

This is read "The intersection of the set of natural numbers and the set consisting of zero, one half, one, three, four and five is the set consisting of one three four and five."

• Example: {1,2,3,7,9} ∪ {0,½,1,3,4,5}= {0,½,1,2,3,4,5,7,9}

This is read "The union of the set consisting of one, two,three,seven and nine and the set consisting of zero,one half, one, three, four and five is the set consisting of zero, one half, one, two, three, four, five, seven and nine."

• Example: D ⊂ C

If C = {1,2,3,4,5,6} and D = {2,4,6} we can say say that "The set D is a subset of the set C."

• Example: D ⊄ E

Or if E = {1,3,5,7} and D = {2,4,6} we would say that "The set D is not a subset of the set E."

Warning:

Computer programmers beware! Do not use the Ø symbol to denote the number zero. While this symbol is often used by computer people to clarify the difference between a zero and the letter "O", math professors recognize Ø only as "the null set". You will be marked down for the inappropriate use of Ø. The number zero should always look like this: 0

Be sure to define your set within curly brackets that look like this: { }. Never use parentheses; your math professor will penalize you harshly for this transgression.