Venn Diagrams and Sets

Moths Butterflies
- Dull Colored - Brightly Colored
- Fly at Night - Fly during Day
- Rest with wings spread - Rest with wings Up
- Fat Furry Bodies - Thin sleek bodies

Things in Common
Moths and Butterflies
both have two wings and
six legs, eat nectar from
flowers, and originally
came from a caterpillar.

The previous picture is called a "Venn Diagram", or "Venn-Euler Diagram".

It shows the similarities and differences between Moths and Butterflies.

The characteristics of moths have been placed in the green circle,
and butterflies in the blue circle.

Because the two insects have some common features,
the two circles have been made to overlap each other.

The common features are put into this common area,
where the two circles cross each other and overlap.

This overlapping region is called the "Intersection" of the Venn Diagram.

The diagram is put inside a big rectangle which is called the "Universal Set".

The "Universal Set" contains everything that we have listed about Moths and Butterflies.

A "Set" is a collection of items that all
belong to a well defined category or group.

For example, a fish belongs to the group of animals
which live in water.

Sets are enclosed in pairs of curly brackets { }

For example, we could write the following for the set of
our favorite animals which live in water:

Water Animals = { Fish, Eel, Platypus }

Some Venn Diagrams Do Not Overlap.

Water Animals = { fish, eel, platypus }

Let’s also have another set of our favorite animals
written like this:

Two Legged Animals = { bird, bat }

When we draw the Venn Diagram for these two sets,
there is nothing in common between the two groups.

(See next slide)

Two Legged Animals Water Animals

Fish
Bird
eel
Bat
Platypus

These two sets are “Mutually Exclusive”. An item belongs in one
circle set, or the other. There are no items that belong to both groups.


Fish
Bird
eel
Bat
Platypus

The “complement” of a set is everything not in that set. (Symbol A’)

The complement of Water Animals is the set:

Not Water Animals = { Bird, Bat }

Let's change our previous example so that it now
contains the following Universal set of Animals:

E = Everything = { Fish, Eels, Platypus,
Penguins, Eagles, Bats }

We are going to use a Venn diagram to divide
these animals into the following two sets:

"Water Animals" and "Two Legged Animals" .

(Continued Next slide)

When we do this, we find that Penguins belong in both
groups:

E = Everything = { Fish, Eels, Platypus,
Penguins, Eagles, Bats }

Water Animals = { Fish, Eels, Platypus, Penguins }

Two Legged Animals = { Penguins, Eagles, Bats }

This means that on our Venn Diagram, we will need to have
two overlapping circles. We then need to put Penguins in
the middle, so that they are inside both circles.
(Continued Next slide)

Fish
Eagles
Eels
Penguins
Bats
Platypus


The union of two sets is everything that is contained within
the two circles all combined together.

** Each item in the Union Set is only listed once **

For our Venn Diagram of Two Legged Animals and Water
Animals, we have:

{ Two Legged Animals } UNION { Water Animals } =

{ Fish, Eels, Platypus, Penguins, Eagles, Bats }

Union is written using a big "U" symbol, or the word "OR".

(See Union Diagram on Next Slide)

The Difference Between "OR" and "AND" is very important.

The word "OR" is confusing for “Union”, because we often
think that our Union set should be everything in one circle
AND everything in the other circle.

“OR” actually means that everything in our Union answer set
is either in one circle, OR in the other circle.

The word "AND" is used for "Intersection“ of sets, and
means that the item is contained in both sets.

Eg. For our previous example, Penguins are in the sets:
{ Two Legged Animals } AND { Water Animals }.

Counts of Items Diagram
There are 3 animals that
Fish live in water, but all 3 of
Eagles
these are not animals that
Eels
Penguins are “Two Legged” .
Bats
Platypus


2 3
Counts of Items Diagram 1
There are 2 animals which
are two legged , but they
do not go into water at all.

Intersection can be written as:
{ Two Legged Animals } Intersection { Water Animals }

{ Two Legged Animals } AND { Water Animals }

{ Two Legged Animals } ∩ { Water Animals }

T ∩ W

Union can be written as:
{ Two Legged Animals } Union { Water Animals }

{ Two Legged Animals } OR { Water Animals }

{ Two Legged Animals } U { Water Animals }

T U W

Complement can be written as:
Complement of { Two Legged Animals } or

{ Two Legged Animals } „ or T„ or NOT in T

Complement of { Water Animals } or

{ Water Animals } „ or W„ or NOT in W

Mutually Exclusive (or “ME”) sets:
Have Nothing in Common
Have No Intersection (Intersection = Null Set or Φ )

We can do Union and combine items in “ME” sets

We can do the Complements of “ME” sets

“ME” sets are also called “Disjoint” sets.

Subsets that are inside other sets:
Have everything in Common

All the items in one group, also belong to the other group

Have an Intersection that is the entire subset

Are drawn as a circle inside a circle

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Venn Diagrams and Sets

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