A _set_, $S$ is a [[mathematical object]] that contains a collection of [[mathematical object]]s. The objects contained in a set are referred to as _[elements](Sets.md#What%20can%20belong%20in%20a%20set)_ of that set. In terms of its elements, a set $S$ is denoted as $S=\{a,b,c,...\}$ where the statement "an object $s$ is an element of a set, $S
quot; is written symbolically as $s\in S$ or $s\in\{a,b,c,...\}.$ Every non-[empty](Sets.md#The%20null%20set) contains elements that belong in that set. ^02f9a3
This definition if taken as complete is part of what's referred to as a _[naive set theory](Sets%20(index).md#Naive%20and%20Axiomatic%20set%20theories)._ However, naive set theory leads to [paradoxes](Sets.md#What%20can%20belong%20in%20a%20set). Thus the notion of sets requires additional [constraints.](Axiomatic%20definition%20of%20a%20set.md)
An object $s$ may be in more than one set, thus sets also serve to categorize mathematical objects. Extending the notion that sets categorize objects, we define [relationships between sets](Sets.md#Relationships%20between%20sets), which give rise to the mathematics of set theory. ^85473b
# [[Cardinality]]
([...see more](Cardinality.md))
# [[Multisets]]
([...see more](Multisets.md))
# [[The null set]]
([...see more](The%20null%20set.md))
# Singletons
A _singleton_ or a _unit set_ is any [set](Sets.md) that contains only 1 element.
Every singleton also gives rise to the [trivial group.](Groups.md#Trivial%20groups) ^2b9ca5
%%There is a claim in Algebra chapter 0 that such a set is necessarily also a group (pg 41.) Try to explain it.%%
# Relationships between sets
There are two ways in which we define relations between sets. These are in terms of the notion that [elements](Sets.md#Sets%20and%20their%20elements) belong in some sets and not others as well as in multiple sets and in terms of the notion that one may define a [rule.](Sets.md#Maps) Below we stick to intuitive definitions that give rise to [naive set theory](Sets%20(index).md#Naive%20and%20Axiomatic%20set%20theories) as well as notation conventions that encode these definitions. %%Does the below content only assume naive set theory or does it stll stand firm in the face of the non-naive approach?%% ^8486ee
## Sets and their elements
Consider a pair of sets $A$ and $B$ where we denote an element of $a$ as $a\in A$ and an element of $B$ as $b\in B.$ ^9e171b
### [[Subsets]]
([...see more](Subsets.md))
### [[Union of sets]]
([...see more](Union%20of%20sets.md))
### [[Intersection of sets]]
([...see more](Intersection%20of%20sets.md))
### [Difference](Set%20difference.md) between sets
([...see more](Set%20difference.md))
### [[Cartesian product]]
([...see more](Cartesian%20product.md))
## Maps
([...see more](Maps.md))
## What can belong in a set
A [set](Sets.md) contains [[mathematical object]]s and since sets are themselves mathematical objects we may have sets containing sets. However there are some constraints to this. In order to avoid [Russell's Paradox](Sets.md#Russell's%20Paradox) we only allow sets of sets if those sets only contain objects of the same [subtype](type.md#subtypes).
[Sets](Sets.md) of elements of more than one [[type]] are allowed.
### Building a set
Becuase it may not be convenient or possble to list every element in a [set,](Sets.md) we may use _set builder notation_ in order to define a set. This pairs elements in a [type](type.md) with a mathematical statement about that type.
Consider a variable $w$ of [type](type.md) $T,$ if we define a statement $P$ that makes a claim about type $T$ then we may define a set $A$ such that $A=\{w|P\}$ where $A$ is the set of things of type $T$ for which $P$ is true^[An alternative notation is $\{w:P\}$.]
If we know already that $w$ belongs in a [set,](Sets.md) $S$ we may define the set $A$ as $A=\{w\in S|P\}$ by indicating belonging with the $\in$ symbol. %%This second way of building sets is slightly circular it seems. Although this type of definition is used in lots of books, according to Aluffi on pg. 2, this style of defintion leads to Russel's paradox.%%
### Russell's Paradox
([...see more](Russell's%20Paradox.md))
### Cantor's Paradox
([...see more](Cantor's%20Paradox))
# Sets as categories
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# Proofs and examples
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# Recommended reading
.md#^e0a7ce)
Thus the following textbooks that are focused on other areas of mathematics contain comprehensive introductions to sets and how we can work with them from the perspective of [naive set theory:](Sets%20(index).md#Naive%20and%20Axiomatic%20set%20theories)
* [Munkres, J. R., _Topology_, Prentice Hall, 2nd Edition, 2000.](Munkres,%20J.%20R.,%20Topology,%20Prentice%20Hall,%202nd%20Edition,%202000..md) pgs. 3-72. This text introduces also some concepts from logic, which is required to make and understand statements about sets.
* [Aluffi, P., _Algebra Chapter 0_, American Mathematical Society, 2009.](Aluffi,%20P.,%20Algebra%20Chapter%200,%20American%20Mathematical%20Society,%202009.md) pgs. 1-16. This section assumes prior knowledge of logic, which is required to make and understand statements about sets.
#MathematicalFoundations/SetTheory