A set is a collection of _unordered_ elements.
A set cannot contain duplicates.
## Notation
### Set Notation
Declaration of a set $A$ with elements $a$, $b$, $c$: 
$$ A := {a, b, c} $$
### Cardinality
Amount of Elements in a set $A$
Notation: $|A|$
$$
A := {1, 2, 3, 4} \
|A| = 4
$$
### Well-Known Sets
- Empty Set: $emptyset = {}$
- Natural Numbers: $N = {1, 2, 3, ...}$
- Integers: $ZZ = {-2, -1, 0, 1, 2}$
- Rational Numbers: $QQ = {1/2, 22/7 }$
- Real Numbers: $RR = {1, pi, sqrt(2)}$
- Complex Numbers: $CC = {i, pi, 1, sqrt(-1)}$

### Set-Builder Notation
Common form of notation to create sets without explicitly specifying elements.
$$
A := {x in N | 0 <= x <= 5} \
A = {1, 2, 3, 4, 5}
$$
### Member of
Denote whether $x$ is an element of the set $A$
Notation: $x in A$
Negation: $x in.not A$

### Subsets
| Type                     | Explanation                                                            | Notation            |
| ------------------------ | ---------------------------------------------------------------------- | ------------------- |
| **Subset**               | Every element of $A$ is in $B$                                         | $A subset B$        |
| **Subset or equal to**   | Every element of $A$ is in $B$, or they are the exactly same set       | $A subset.eq B$     |
| **Proper subset**        | Every element of $A$ is in $B$, but $A$ is definitely smaller than $B$ | $A subset.sq B$<br> |
| **Superset**<br>         | $A$ contains everything that is in $B$                                 | $A supset B$        |
| **Superset or equal to** | $A$ contains everything that is in $B$, or they are identical          | $A supset.eq B$     |

## Operations
### Union
Notation: $A union B$
Definition: all elements from both sets _without adding duplicates_
$$ A := {1, 2, 3}\ B := {3, 4, 5}\ A union B = {1, 2, 3, 4, 5} $$
### Intersection
Notation:$A inter B$
Definition: all elements _contained in both sets_
$$
A := {1, 2, 3} \
B := {2, 3, 4} \
A inter B = {2, 3}
$$
### Difference
Notation: $A backslash B$ 
Definition: all elements _in $A$ that are not in $B$_
$$
A := {1, 2, 3} \
B := {3, 4, 5} \
A backslash B = {1, 2}
$$ 
### Symmetric Difference
Notation: $A Delta B$ 
Definition: all elements _only in $A$ or only in $B$_
$$
A := {1, 2, 3} \
B := {2, 3, 4} \
A Delta B = {1, 4}
$$
### Cartesian Product
Notation: $A times B$
Definition: all pairs of all elements in $A$ and $B$
$$
A := {1, 2} \
B := {3, 4, 5} \
A times B = {(1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5)}
$$
### Powerset
Notation: $cal(P)(A)$
Definition: all possible _Subsets of A_
$$
A := {1, 2, 3} \
cal(P)(A) = {emptyset, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}
$$

## Closures
A **closed set** under the binary operation $*$ denoted by $forall a,b in S, a * b in S$
"All $a * b$ from set $S$ are included in set $S$"

A set can be closed under any binary operation ($+, -, *, \/$), depending on the set itself if it actually is closed or not. $NN$ for example is not closed under subtraction.
