How To Find The Universal Set In A Venn Diagram

Sets and Venn Diagrams

Sets

A set is a collection of things.

For example, the items you habiliment is a set: these include hat, shirt, jacket, pants, and so on.

You write sets within curly brackets similar this:

{chapeau, shirt, jacket, pants, ...}

You can also have sets of numbers:

• Set of whole numbers: {0, 1, 2, 3, ...}
• Set of prime numbers: {2, three, five, vii, eleven, thirteen, 17, ...}

X Best Friends

You lot could have a fix made upwardly 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. Information technology is normal to use lowercase messages for them.

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

Soccer = {alex, casey, drew, hunter}

(It says 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}

We can put their names in two carve up circles:

Marriage

You tin now listing 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 ... merely your friends that play Soccer or Lawn tennis (or both).

In other words we combine the elements of the 2 sets.

Nosotros tin can show that in a "Venn Diagram":

Venn Diagram: Union of 2 Sets

A Venn Diagram is clever because information technology shows lots of information:

• Practise you see that alex, casey, drew and hunter are in the "Soccer" set?
• And that casey, drew and jade are in the "Lawn tennis" set?
• And hither is the clever thing: casey and drew are in BOTH sets!

All that in one small diagram.

Intersection

"Intersection" is when you must be in BOTH sets.

In our instance 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 nosotros write it:

Soccer ∩ Lawn tennis = {casey, drew}

In a Venn Diagram:

Venn Diagram: Intersection of 2 Sets

Which Style Does That "U" Get?

Think of them as "cups": ∪ holds more water than ∩, right?

And so Marriage ∪ is the one with more elements than Intersection ∩

Divergence

Y'all can also "subtract" one set up 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 information technology:

Soccer − Tennis = {alex, hunter}

In a Venn Diagram:

Venn Diagram: Deviation of 2 Sets

Summary So Far

• ∪ is Wedlock: is in either set or both sets
• ∩ is Intersection: only in both sets
• − is Difference: in ane set but not the other

Iii Sets

You can also use Venn Diagrams for 3 sets.

Let the states say the tertiary set is "Volleyball", which drew, glen and jade play:

Volleyball = {drew, glen, jade}

But let'south be more "mathematical" and utilise a Upper-case letter Letter of the alphabet for each set:

• South means the prepare of Soccer players
• T means the set of Tennis players
• V means the set up of Volleyball players

The Venn Diagram is now similar this:

Union of 3 Sets: Southward ∪ T ∪ 5

Y'all 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 simply Lawn tennis

Nosotros can at present have some fun with Unions and Intersections ...

This is but the set S

South = {alex, casey, drew, hunter}

This is the Union of Sets T and V

T ∪ V = {casey, drew, jade, glen}

This is the Intersection of Sets Due south and V

S ∩ V = {drew}

And how about this ...

• take the previous fix S ∩ Five
• and so subtract T:

This is the Intersection of Sets S and V minus Fix T

(South ∩ V) − T = {}

Hey, in that location is nothing at that place!

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

The Empty Set has no elements: {}

Universal Gear up

The Universal Gear up is the prepare that has 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 Spousal relationship. You just take to exist conscientious, OK?

In our case the Universal Set is our 10 Best Friends.

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

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

Now you tin can encounter ALL your x all-time friends, neatly sorted into what sport they play (or not!).

And then we tin do interesting things like accept the whole set up and subtract the ones who play Soccer:

Nosotros write it this fashion:

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

Which says "The Universal Set minus the Soccer Set is the Prepare {blair, erin, francis, glen, ira, jade}"

In other words "everyone who does not play Soccer".

Complement

And there is a special fashion of saying "everything that is not", and it is called "complement" .

We show information technology by writing a little "C" similar this:

S^c

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

Southward^c = {blair, erin, francis, glen, ira, jade}
(exactly the same as the U − Due south example from above)

Summary

• ∪ is Wedlock: is in either set or both sets
• ∩ is Intersection: only in both sets
• − is Difference: in 1 set up but non the other
• A^c is the Complement of A: everything that is not in A
• Empty Set up: the prepare with no elements. Shown past {}
• Universal Set: all things nosotros are interested in

1886, 1887, 1890, 1892, 7220, 1888, 1889, 1891, 91, 73

Source: https://www.mathsisfun.com/sets/venn-diagrams.html

Share this post