# Set theory
## Basic Definition

The term "set" can be thought as a well-defined collection of objects. In set theory, These objects are often called "elements".
* We usually use capital letters for the sets, and lowercase letters for the elements.
* If an element $a$ belongs to a set $A$, we can say that "$a$ is a member of the set $A$", or that "$a$ is in $A$", or simply write $a\in A$.
* Similarly, if $a$ is not in $A$, we would write $a\notin A$.
(Example) $x\in \mathbb {R}$. In this case, $x$ is the element and $\mathbb {R}$ is the set of all real numbers.

### Notation

Example of a common notation style for the definition of a set:
* $S$ is a set
* $P(x)$ is a property (The elements of $S$ may or may not satisfy this property)
* Set $A$ can be defined by writing:
$A=\{x\in S\mid P(x)\}$
This would read as "the set of all $x$ in $S$, such that $P$ of $x$."

### Elements

There are two ways that we could show which elements are members of a set: by listing all the elements, or by specifying a rule which leaves no room for misinterpretation. In both ways we will use curly braces to enclose the elements of a set. Say we have a set $A$ that contains all the positive integers that are smaller than ten. In this case we would write $A=\{1,2,3,4,5,6,7,8,9\}$. We could also use a rule to show the elements of this set, as in $A=\{a:{\text{ }}a{\text{ positive integer less than 10}}\}$.
In a set, the order of the elements is irrelevant, as is the possibility of duplicate elements. For example, we write $X=\{1,2,3\}$ to denote a set $X$ containing the numbers 1, 2 and 3. Or,$X=\{1,2,3\}=\{3,2,1\}=\{1,1,2,3,3\}$.

## Subsets

Formal universal conditional statement: "set A is a subset of a set B"
* $A\subseteq B\Leftrightarrow \forall x$, if $x\in A$, then $x\in B.$
Negation:
* $A\nsubseteq B\Leftrightarrow \exists x$ such that $x\in A$ and $x\notin B.$
Truth Table Example:
| $x\in A$ | $x\in B.$ | if $x\in A$, then $x\in B.$ | $x\in A$ and $x\notin B$ |
| 1 | 1 | 1 | 0 |
| 1 | 0 | 0 | 1 |
| 0 | 1 | 1 | 0 |
| 0 | 0 | 1 | 0 |
A proper subset of a set is a subset that is not equal to its containing set. Thus
A is a proper subset of B $\iff$

### Set Identities

Let all sets referred to below be subsets of a universal set U.
(a) A ∪ ∅ = A and (b) A ∩ U = A.
5. Complement Laws:
(a) A ∪ A c = U and (b) A ∩ A c = ∅.
6. Double Complement Law:
(A c ) c = A.
7. Idempotent Laws:
(a) A ∪ A = A and (b) A ∩ A = A.
8. Universal Bound Laws:
(a) A ∪ U = U and
(b) A ∩ ∅ = ∅.
| $A\cap U=A$ | $A\cap U=A$ |
| Identity | For all sets A | |
| Identity Laws: | $A\cup \emptyset =A$ | $A\cap U=A$ $A\cap U=A$ |
| Complement Laws: | $A\cup A^{c}=U$ $A\cap A^{c}=\emptyset$ | |
| Double Complement Law: | $\left(A^{c}\right)^{c}=A$ | |
| Idempotent Laws: | $A\cup A=A$ $A\cap A=A$ | |
| Universal Bound Laws: | | |
| Identity | For all sets A and B |
| Commutative Laws: | $A\cup B=B\cup A$$A\cap B=B\cap A$ |
2. Associative Laws: For all sets A, B, and C,
(a) (A ∪ B) ∪ C = A ∪ (B ∪ C) and
(b) (A ∩ B) ∩ C = A ∩ (B ∩ C).
3. Distributive Laws: For all sets, A, B, and C,
(a) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) and
(b) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).
For all sets A and B,
De Morgan’s Laws:
(a) (A ∪ B) c = A c ∩ B c and (b) (A ∩ B) c = A c ∪ B c .
Absorption Laws:
(a) A ∪ (A ∩ B) = A and (b) A ∩ (A ∪ B) = A.
Set Difference Law:
* A − B = A ∩ B c .
Complements of U and ∅:
* $U^{c}=\emptyset$
* $\emptyset ^{c}=U$

## Cardinality

The cardinality of a set is the number of elements in the set. The cardinality of a set $A$ is denoted $|A|$.

### Types of Sets by Cardinality

A set can be classified as finite, countable, or uncountable.
* Finite Sets are sets that have finitely many elements, $A=\{1,2,3\}$ is a finite set of cardinality 3. More formally, a set $A$ is finite if a bijection exists between $A$ and a set $\{1,\ldots ,n\}$ for some natural number $n$. $n$ is the said set's cardinality.
* Countable Sets are sets that have as many elements as the set of natural numbers. As since $|\mathbb {Q} |=|\mathbb {N} |$, the set of rational numbers is countable.
* Uncountable Sets are sets that have more elements than the set of natural numbers. As since $|\mathbb {R} |>|\mathbb {N} |$, the set of real numbers is uncountable.

### Common Sets of Numbers

* $\mathbb {N}$ is the set of Naturals
* $\mathbb {Z}$ is the set of Integers
* $\mathbb {Q}$ is the set of Rationals
* $\mathbb {R}$ is the set of Reals
* $\mathbb {C}$ is the set of Complex Numbers

### Comparison of Sets

2 sets $A$ and $B$ have the same cardinality (i.e. $|A|=|B|$), if there exists a bijection from $A$ to $B$. In the case of $|\mathbb {Q} |=|\mathbb {N} |$, they are the same cardinality as there exists a bijection from $\mathbb {Q}$ to $\mathbb {N}$.

## Partitions of Sets

In many applications of set theory, sets are divided up into non-overlapping (or disjoint) pieces. Such a division is called a partition.
Two sets are called disjoint if, and only if, they have no elements in common.
   A and B are disjoint ⇔ A ∩ B = ∅.
Sets $A_{1},A_{2},A_{3},\ldots$ are mutually disjoint (or pairwise disjoint or nonoverlapping)
if, and only if, no two sets $A_{i}$ and $A_{j}$ with distinct subscripts have any elements in
common. More precisely, for all $i,j=1,2,3,\ldots$
$A_{i}\cap A_{j}=\emptyset$ whenever $i\not =j$.

## Power Sets

The power set of a set A is all possible subsets of A, including A itself and the empty set. Which can be represented: $P_{(A)}=\{\emptyset ,\{1\},\{2\},\ldots \}$
For the set $A=\{1,2,3\}$
$P_{(A)}=\{\emptyset ,\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\}\}$

## Learning resources
### Wikiversity

* Introduction_to_set_theory/Lecture_1

### Wikipedia

* Set theory category