Wolfram Alpha:

```Set Theory
----------

ℙ = prime numbers

ℝ = real numbers

ℤ = integers

ℂ = complex numbers

ℕ = natual numbers

ℚ = rationals (fractions)

Subsets
-------

Number of subsets = 2n

Every set contains the subset, ∅ = {}

Example:

{40,50} has 4 subsets:  P = {40,50}, Q = {40}, R = {50}, ∅ = {}

P ⊆ {40,50}, Q ⊂ {40,50}, R ⊂ {40,50} and ∅ ⊆ {40,50}

⊆ means subset or equal.  ⊂ means proper subset.  A proper
subset is defined as:

If A ⊆ B and A ≠ B then it is a proper subset.

Note that the null set is NOT a proper subset.

Set Notation
------------

{x ∈ Z| x = 2m + 1, m ε Z}

Is a fancy way of saying "all integers x such that x is equal
to 2 times m plus 1, where m is an integer"

{x|x ∈ Z, x < 10}

Means "all integers x that are less than 10".

Venn Diagrams
-------------

2 sets:

Universal set:  {4}
∪ ≡ UNION ≡ OR
∩ ≡ INTERSECTION ≡ AND
~ ≡ NOT ≡ COMPLEMENT

A ∪ B:     {1,2,3}
A ∩ B:     {2}
~A:        {3,4}
A - B:     {1}
~(A ∪ B):  {4}
~(A ∩ B):  {1,3,4}

3 sets:

A ∪ B:  {1,2,4,5,6,7}
A ∩ B:  {4,5}
A ∪ B ∪ C:  {1,2,3,4,5,6,7}
A ∩ B ∩ C:  {5}
~A:  {3,6,7,8}
A - B - C:     {1}
~(A ∪ B ∪ C):  {8}
~(A ∩ B ∩ C):  {1,2,3,4,6,7}
A ∪ (B ∩ C):  {1,2,4,5,6}
A ∩ (B ∪ C):  {2,4,5}
and so on...

Construction
------------

To construct a Venn Diagram corresponding to sets A, B and C
we use the following procedure:

1.  Compute A ∩ B ∩ C

2.  Compute A ∩ B

3.  Compute B ∩ C

4.  Compute A ∩ C

So for,

A = {1,2,3,4}  B = {1,3,4,6,8}  C = {1,3,6,9}

A ∩ B ∩ C = {1,3}

A ∩ B = {1,3,4}

B ∩ C = {1,3,6}

A ∩ C = {1,3}

Which looks:

Example:

There 24 flute, piano and guitar players.

10 play the flute, 14 play the piano and 13 play the guitar.

2 only play the flute, 5 only play the piano and 7 only play the guitar.

1 plays flute AND guitar but NOT piano.

2 play piano AND guitar but NOT flute.

3 play piano AND guitar AND flute.

How many people play the piano and the flute but not the guitar?

P ∩ F ∩ ~G = 14 - 5 - 2 - 3 = 4

Note:  Sum of all combinations has to equal 24.

Probability = 4/24 = 1/6

```