1. Set Notation Subset Element Union Intersection Not a Subset
Null or Empty Set
Not an Element

2. Set Notation A set, informally, is a collection of things.
The "things" in the set are called the "elements", and are listed inside curly braces.
Lets construct : "the set of things in our classroom.” What are the elements we want to include in our set?
How do we write our set?

3. Naming Sets-Use Capital Letters
Sets are usually named using capital letters. So let's name this set as "A". Then we have:
A = {desk, chair, table, file cabinet, bookcase}

4. We use a special character to say that something is an element of a set. It looks like a funny curvy capital E.
          
“desk is an element of the set A”

5. Sets in Math {x is a natural number, x < 10}
The elements of a set can be listed out according to a rule, such as:
{x is a natural number, x < 10}
Technically, you should use full "set-builder notation", which looks like this:
                
"the set of all x,
such that x is an element of the 
natural numbers less than 10"

6. 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, is called using the “roster method".

7. Common Symbols
In using set notation, you may see these common symbols used:
N: THE COUNTING NUMBERS
Z: THE INTEGERS
Q: The Fractions
R: The real numbers

8. Let’s Try a Few:
Write the following sets using set notation and the roster method:
The integers between -5 and +5

9. So you try a few
Write the following sets using set notation and the roster method:
The set of integers larger than or equal to 18
{ 18, 19, 20, 21,…}

10. Subsets
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:

11. Not a Subset
To show something is not a subset, you draw a slash through the subset symbol. Again, suppose
A = {1, 2, 3} and
B = {1, 2, 3, 4, 5, 6}.
This is pronounced as "B is not a subset of A".

12. Unions
If two sets are being combined, this is called the "union" of the sets, and is indicated by a large U-type character. So if C = {1, 2, 3, 4, 5, 6} and D = {4, 5, 6, 7, 8, 9}, then

13. Intersections
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:

14. Try:
Let A = {red, orange, ball, sock} and Let B = {1, 2, 3, red, sock} Find

15. Try:
Let A = and Let B= {1, 3, 5, 7, 9, 11, 13} Find

16. Complements
universal set is a set containing all elements of a problem under consideration
A complement is the set of all elements from the universal set that are not part of the mentioned set.
For example:
universal set is {1, 2, 3, 4, 5} A ={1, 2, 3}
the complement of set A would be {4,5}
In set notation, complements can be shown using three different symbols:
~A, Ac, and A′ all mean the complement of set A.

17. If the universal set is all integers, and A={ …-2, 0, 2, 4, 6…} find A′

18. Null Sets
If you are asked for the intersection of two disjoint sets, there will be no common elements. This solution set will be called the “null set” and written as ” Ø . This symbol is written alone and without brackets.
There are no numbers that are both even and odd. Therefore, these sets are disjoint sets. No elements in common.
Even
Numbers
Odd
Numbers

19. Examples of Null Sets A = {1, 2, 3} B = {4, 5, 6} A ∩ B = Ø
A = {All the boys in 8th grade} B = {All the principals at your school}