A union of two or more sets is another set that contains everything contained in the previous sets.

Union is designated by the symbol È.
If A and B are sets then A È B represents the union of A and B

In the examples below you'll see the union of sets, then it will be time for you to put together some sets of your own.

Example 1
A={1,2,3,4,5} B={5,7,9,11,13}
A È B = {1,2,3,4,5,7,9,11,13}
Notice that when I wrote out the united set I did not write "5" twice. I simply listed all of the new sets elements.

Example 2
A={all the books written by Charles Dickens}
B={all the books written by Mark Twain}
A È B = {all books written by either Charles Dickens or Mark Twain}

• In order for the computer to check your answer the entries must be in numerical (or alphabetical) order.
  This is generally not true for sets.
• Place commas between the elements
• Your answer should start with a "{" and end with a "}"

The intersection of two (or more) sets is those elements that they have in common.
Intersection is designated by the symbol Ç.
So if A and B are sets then the intersection (the elements they both have in common) is donated by A Ç B.

In the following examples you will have a chance to see intersection in action.

Example 1
A={1,3,5,7,9} B={2,3,4,5,6}
The elements they have in common are 3 and 5
A Ç B = {3,5}

Example 2
A={The English alphabet} B={vowels}
So A Ç B = {vowels}

Example 3
A={1,2,3,4,5} B={6,7,8,9,10}
In this case A and B have nothing in common.
As usual, we have a symbol for this phenomenon: {Æ}.
{Æ} is called the “empty set.”
A Ç B = {Æ}
(See back on page 1 of this unit, I told you that there would be a set with nothing in it)

• In order for the computer to check your answer the entries must be in numerical (or alphabetical) order.
  This is generally not true for sets.
• Place commas between the elements
• Your answer should start with a "{" and end with a "}"

However if even one element of one set is not contained within the other then thy are not subsets.
If A were defined as {1,2,3,4,5} and B as {3,4,5,6} then B would not be a subset of A since
“6” Î B but 6 Ï A.
The symbol for “not a subset” is Ë.
We would write B Ë A.

The notion of subsets is graphically illustrated below

Finally we’ll look at two more examples, and then I’ll make one final point

Example 1
A={1,2,3,4,5,6,7} B={2,3,4}
B is entirely within A (i.e. every element of B is also an element of A) so we can write B Ì A

Example 2
A={1,2,3,4,5} B={2,4,6}
Here B Ë A

One last point I need to make then you can work out some examples regarding subsets.
As in the rest of life everything is a matter of perspective.
“A is a subset of B” could also be viewed as “B contains A.”
This notion of "contain" is symbolized by É.
So A Ì B is the same thing as B É A.

If A Ì B and B Ì C then A Ì C

If A Ì B then A Ç B = A.

If A Ì B then A È B = B.

Go to the top of this page
Go to the set home page
Go to the next page of this unit