Topic 10: Sets and Venn Diagrams

Ten Best Friends

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

You can now list your friends that play Soccer OR Tennis.

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 your friends that play Soccer, Tennis or both).

We can also put it in a "Venn Diagram":

A Venn Diagram is clever because it 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:

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:

Summary So Far

• ∪ is Union: is in either set
• ∩ is Intersection: must be in both sets
• − is Difference: in one set but not the other

Three Sets

You can also use Venn Diagrams for 3 sets.

Let us say the third set is "Volleyball" which is drew, glen and jade play:

Volleyball = { drew, glen, jade }

But let's be more "mathematical" and use a Capital Letter for each set:

• S means the set of Soccer players
• T means the set of Tennis players
• V means the set of Volleyball players

The Venn Diagram is now like this:

You can see (for example) that:

• drew plays Soccer, Tennis and Volleyball
• jade plays Tennis and Volleyball
• alex and hunter play Soccer, but don't play Tennis or Volleyball
• no-one plays only Tennis

We can now have some fun with Unions and Intersections.

S = { alex, casey, drew, hunter }

T ∪ V = { casey, drew, jade, glen }

S ∩ V = { drew }

And how about this

• take the previous set S ∩ V
• then subtract T:

(S ∩ V) − T = {}

Hey, there is nothing there!

That is OK, it is just the "Empty Set". It is still a set, so we use the curly brackets with nothing inside: {}

The Empty Set has no elements: {}