Set

A set is the mathematical model for a collection of different things; a set contains elements (so a collection of elements). 
We often think about sets as lying in some universal set (often denoted \Omega, S, \mathbb{U}, or U), which can be thought of as a collection that contains all the entities one wishes to consider in a given situation. 
==Notation==
A set is usually denoted by a capital letter, e.g. A. If an element a is in the set we write a \in A or a \notin A if not in the set. 
[Q: WOULD IT BE RIGHT TO SAY THAT ELEMENT, SUBSET, AND SET EQUALITY ARE BINARY RELATIONS; EMPTY SET, DISJOINT AND CLASS ARE DEFINITIONS, AND COMPLEMENT, UNION, ETC. ARE BINARY OPERATIONS?] 
There are generally three methods to denote (and possibly define sets). 
#Roster (a.k.a. enumeration) notation: specifies a set by a list of elements in curly brackets, e.g. \{a, b, c\}. If there are many elements with an implicit pattern, often denoted using triple colons as an abbreviation as e.g. \{a_1, a_2, ..., a_N\}, or for infinite sets by \{a_1, a_2, ...\}.
#Semantic definition: specifies a set by a 'rule' that all elements satisfy. E.g. let A be the set of all even numbers.
#Set-builder notation: specifies a set as a selection from a larger set. E.g. F = \{x \in \mathbb{R} \mid \text{x is an integer}\}.
==Subset and Superset ==
*A is a subset of B if a every element in A is also in B, A \subseteq B \iff \forall x \in A, x \in B 
*A is a superset of B if every element in B is also in A, A \supseteq B \iff \forall x \in B, x \in A
Hence, if A is a subset of B, then B is a superset of A.
== Set Equality ==
Two sets are equal, A = B, if they contain exactly the same elements. That is, if \forall x \in A,~ x \in B and \forall y \in B,~ y \in A. 
* Note: it follows from this definition that \{2,4,6\} = \{2,6,4\}. Hence, when writing a set in roster notation, the order does not matter.
* Note: it follows from this definition that \{2,4,4\} = \{2,4\}. Hence, when writing a set in roster notation, repeated elements do not change the set. E.g. the set of letters in the word Mississippi is equal to the set \{M,i,s,p\}.
* Hence, it follows from the definition of set equality that \{a, 1, 1\} = \{1, a\}.
* Note: a theorem follows from the definitions of subset and set equality : A = B \iff A \subseteq B ~\text{and}~ B \subseteq A. (note, this is a theorem as it is not a definition itself, but follows from two definitions)
Alternatively, may want to rewrite this part as; properties of set equality and add their proofs from the definition of set equality.
* \{x\} = \{x,x\}
* \{x,y\} = \{y,x\}
* \{x\} = \{y,z\} \iff x = y = z
== Empty Set==
The empty set is a set with no elements in it, and is denoted as \varnothing = \{\}.In Latex: 
\varnothing
==Class/Set of Sets==
A set whose elements are sets (a set of sets) is called a class, family, collection, or set system. Often, a family is denoted with a calligraphic letter like A_1, A_2 \in \mathcal{A} where A_1, A_2 are sets (and \mathcal{A} \subseteq 2^\Omega where \Omega is the universal set). 
Some special families/classes of sets, when addition structure is added (see Space (math)), have special names, e.g. pi-systems, lambda-systems, algebras, sigma-algebras, and topologies. 
== Disjoint Sets ==
* Disjoint: If the intersection of two sets is empty A \cap B = \varnothing, then the two sets are disjoint.
* Pairwise Disjoint: a family of sets is pairwise disjoint if any two sets in it are disjoint; if for any A_i, A_j \in (A_i)_{i \in I}wheneveri \neq j, A_i \cap A_j = \varnothing. 
Note that if two sets are disjoint, then A \setminus B = A and A \triangle B = A \cup B.
== Power Set==
The power set of a set A is the set of all subsets of A, including the empty set and A itself, denoted by 2^{A}.Sometimes also denoted as P(A), \mathbb{P}(A), \mathcal{P}(A), etc. Hence, the power set is a family/class.
The power set can not be empty, by definition it contains at least two elements, \varnothing and A. 
== Basic Set Operations==
For more results, e.g. De Morgan’s laws, see Algebra of Sets. Assuming the sets that we study live in some larger set \Omega, the following basic operations are defined:
*Complement: A^c = \{x \in \Omega \mid x \notin A \}, the set of elements not in a set.
*Intersection: A \cap B = \{x \in \Omega \mid x \in A \text{ and } x \in B\}, the set of elements that are both in set A and set B.
*Union: A \cup B := \{x \in \Omega \mid x \in A \text{ or } x \in B\}, the set of elements in A, B, or both.
*Difference: A \setminus B := \{x \in A \mid x \notin B \} = A \cap B^{c}, the elements in A that are not in B.
*Symmetric Difference: A \triangle B := (A \cup B) \setminus (A \cap B), the elements inA or B, but not in both.
*Cartesian Product: A \times B := \{ (x, y) \mid x \in A \text{ and } y \in B \}, this is a way of defining ordered pairs/sequences/lists from sets. [ADD AND LINK TO ARTICLE WITH DETAILS ON OPERATIONS ON CARTESIAN PRODUCT].
==Important Sets of Numbers==
Some sets are referred to more often, and have been given their own notation. These include:
*\mathbb{N}: \{1, 2, 3, ...\}, the 'natural' or 'counting' numbers (conventions differ on whether zero is included).
*\mathbb{Z}: \{..,-2, -1, 0, 1, 2,...\}, the set of integers
*\mathbb{Q}: the set of all rational numbers (that is, the set of all proper and improper fractions)add link to fractions page for the different types
*\mathbb{R}: the set of all real numbers
*\mathbb{C}: the set of all complex numbers, \{a +ib ~|~ a, b \in \mathbb{R}\}
*\mathbb{R}^n: the set of all real-valued n \times 1 dimensional vectors
*\mathcal{C}(A,B): the set of all continuous functions from set A to set B.
Clearly, \mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}.