sets n venn diagram

                                                      Sets                                          

A collection of things.
For example, the items you wear is a set: these would include shoes, socks, hat, shirt, pants, and so on.
You write sets inside curly brackets like this:
{socks, shoes, pants, watches, shirts, ...}
You can also have sets of numbers:

• Set of whole numbers: {0, 1, 2, 3, ...}
• Set of prime numbers: {2, 3, 5, 7, 11, 13, 17, ...}

You could have a set made up of your ten best friends:

• {alex, blair, casey, drew, erin, francis, glen, hunter, ira, jade}

Each friend is an "element" (or "member") of the set (it is normal to use lowercase letters for them.)


Now let's say that alex, casey, drew and hunter play Soccer:

Soccer = {alex, casey, drew, hunter}

(The Set "Soccer" is made up of the elements alex, casey, drew and hunter).

And casey, drew and jade play Tennis:

Tennis = {casey, drew, jade}

You could put their names in two separate circles:

                                            Union

This is called a "Union" of sets and has the special symbol ∪:

Soccer ∪ Tennis = {alex, casey, drew, hunter, jade}

Not everyone is in that set ... only certain person that play Soccer or Tennis (or both).
We can also put it in a "Venn Diagram":

Venn Diagram: Union of 2 Sets

A Venn Diagram shows lots of information:

• Do you see that alex, casey, drew and hunter are in the "Soccer" set?
• And that casey, drew and jade are in the "Tennis" set?
• And here is the clever thing: casey and drew are in BOTH sets!

                                                        

                                                    

                                    Intersection

"Intersection" is when you have to be in BOTH sets.
In our case that means they play both Soccer AND Tennis ... which is casey and drew.
The special symbol for Intersection is an upside down "U" like this: ∩
And this is how we write it down:

Soccer ∩ Tennis = {casey, drew}

In a Venn Diagram:

Venn Diagram: Intersection of 2 Sets 

                                     Difference

You can also "subtract" one set from another.
For example, taking Soccer and subtracting Tennis means people that play Soccer but NOT Tennis ... which is alex and hunter.
And this is how we write it down:

Soccer − Tennis = {alex, hunter}

In a Venn Diagram:

Venn Diagram: Difference of 2 Sets 

                                    Universal Set

The Universal Set is the set that contains everything. Well, not exactly everything. Everything that we are interested in now.

Sadly, the symbol is the letter "U" ... which is easy to confuse with the ∪ for Union. You just have to be careful, OK?

In our case the Universal Set is our Ten Best Friends.

U = {alex, blair, casey, drew, erin, francis, glen, hunter, ira, jade}

I can show the Universal Set in a Venn Diagram by putting a box around the whole thing:

Now you can see ALL our ten best friends, neatly sorted into what sport they play (or not!).
And then we can do interesting things like take the whole set and subtract the ones who play Soccer:

I write it this way:

U − S = {blair, erin, francis, glen, ira, jade}

Which says "The Universal Set minus the Soccer Set is the Set {blair, erin, francis, glen, ira, jade}"
In other words "everyone who does not play Soccer".

                                        Complement

And there is a special way of saying "everything that is not", and it is called "complement".
We show it by writing a little "C" like this:

S^c

Which means "everything that is NOT in S", like this:

S^c = {blair, erin, francis, glen, ira, jade}
(just like the U − C example from above)

Summary of the Sets Notation

∪ is Union: ALL elements in BOTH sets
∩ is Intersection: elements where sets overlap      
− is Difference: in one set but not the other
A^c is the Complement of A: elements NOT in the set
Empty Set: the set with no elements. Shown by {}
Universal Set: all things we are interested in 

 

 HERE A VIDEOS OF EXPLANATION: