# Sets: Summary

## Sets: Introduction

* German mathematician Georg Cantor (1845-1918) has introduced sets. His work from 1874 to 1884 is the origin of set theory. Cantor's definition: "*A set is a gathering together into a whole of definite distinct objects of our perception or thought, called the set elements*".
* Sets are ubiquitous in mathematics, with many math objects resabling a set of some kind.
* Sets are introduced in theories without a proof: as a primitive or axiom.
* Zermelo-Fraenkel (ZF) set theory is considered the foundation of modern math.
* Initially, naïve set theory was developed that treated sets as collections of objects without imposing any restriction as to what can consitute a set. This has led to some paradoxes, with Russell's paradox being the most famous.
* Naïve set theory is an informal approach to set theory.
* Axiomatic set theory, such as ZF, is a formal system based on a set of axioms.

## Sets: Elements

* objects that belong to a set are called set elements or set members
* relation between an object and a set is a *membership relation*, denoted by a variant of Greek letter epsilon (ε): $$a\in A$$

## Sets: inclusion

* subset is an example of *inclusion relation*
* inclusion relation is reflexive, transitive, and anti-symmetric
* proper inclusion relation is irreflexive, transitive, and asymmetric.&#x20;
* inclusion and proper inclusion relations have natural converses;

  A (properly) includes B iff B is (properly) included in A;

  alternatively, A is a (proper) superset of B
* exclusion relation is symmetric and anti-reflexive
* inclusion relation is antisymmetric
* set membership (belonging) is an example of membership relation
* membership (elementhood) relation is reflexive
* inclusion relation is transitive, membership relation is not

## Sets: Cardinality

* Cardinality of the set X is denoted by $$|X| = n$$
* Cardinality of the powerset of the set X is $$|\mathcal{P}(X)| = 2^{|X|} = 2^n$$
* Cardinality of the Cartasian product of the set X: $$|X \times X| = |X^2| = n^2$$
* Cardinality of the powerset of the Cartasian product of the set X is $$|\mathcal{P}(X^2)| = 2^{(n^2)}$$
* Each relation is a subset of the Cartasian product of the set X: $$R \subseteq X^2$$
* Each relation is an element in the powerset of the Cartasian product of the set X: $$R \in P(X^2)$$
* Number of all possible relations on a $$X^2$$ set: $$|\mathcal{P}(X^2)| = 2^{(n^2)}$$
* standard set notation: curly braces for listing the elements explicitly
* comprehensions: `{x ∈ S |...}`
* `∅` for the empty set
* `S \ T` for the set difference of `S` and `T`
* cardinality of a set `S` is `|S|`
* powerset of `S` i.e. the set of all the subsets of `S`, is `P(S)`
* the set `{0,1,2,3,4,5,...}` of natural numbers is denoted by the symbol `N`
* a set is countable if its elements can be placed in one-to-one correspondence with the natural numbers

## Relations

* An n-place relation on a collection of sets `S1,S2,...,Sn`

  is a set `R ⊆ S1 × S2 ×...× Sn` of tuples of elements `S1 - Sn`
* the elements `s1 ∈ S1` through `sn ∈ Sn` are related by `R` if `(s1,...,sn)` is an element of `R`
* A one-place relation on a set `S` is called a predicate on `S`
* P is true of an element `s ∈ S` if `s ∈ P`
* we write `P(s)` instead of `s ∈ P`, regarding `P` as a function mapping elements of `S` to truth values
* A two-place relation `R` on sets `S` and `T` is a binary relation
* We write `sRt` instead of `(s, t) ∈ R`
* if `S` and `T` are the same set `U`, then `R` is a binary relation on `U`
* 3-place or more place relations are often written using a mixfix concrete syntax, where the elements in the relation are separated by a sequence of symbols that jointly constitute the name of the relation. For example, for the typing relation for the simply typed lambda calculus, we write `Γ ⊢ s : T` to mean the triple `(Γ,s,T)` is in the typing relation.
* The domain of a relation R on sets S and T, written `dom(R)`, is the set of elements `s ∈ S` such that `(s, t) ∈ R` for some `t`.
* The codomain or range of `R`, written `range(R)`, is the set of elements `t ∈ T` such that `(s, t) ∈ R` for some s

## Functions

* A relation R on sets S and T is called a partial function from S to T if, whenever `(s, t1) ∈ R` and `(s, t2) ∈ R`, we have `t1 = t2`.
* If, in addition, `dom(R) = S`, then R is called a total function (or just function) from S to T.&#x20;
* A partial function R from S to T is said to be defined on an argument `s ∈ S` if `s ∈ dom(R)`, and undefined otherwise.
* We write `f (x) ↑`, or `f (x) =↑`, to mean *f is undefined on x*, and `f (x)↓` to mean *f is defined on x*
* we also need to define functions that may fail on some inputs, so it is important to distinguish failure (which is a legitimate, observable result) from divergence
* a function that may fail can be either partial, i.e. it may also diverge, or total (it must always return a result or explicitly fail)
* We write `f(x)=fail` when f returns a failure result on the input x.
* Formally, a function from S to T that may also fail is actually a function

  from S to `T ∪ {fail}`, where we assume that `fail` does not belong to T.
* Suppose R is a binary relation on a set S and P is a predicate on S. P is preserved by R if whenever we have `sRs'` and `P(s)`, we also have `P(s0)`.

## Ordered Sets

* a binary relation R on a set S is reflexive if R relates every element of S to itself i.e. `sRs` or `(s,s) ∈ R` for all `s ∈ S`.
* R is symmetric if `sRt` implies `tRs`, for all s and t in S.
* R is transitive if `sRt` AND `tRu` imply `sRu`.
* R is antisymmetric if `sRt` and `tRs` imply that `s = t`
* A reflexive and transitive relation R on a set S is called a preorder on S.
* We denote a preorder by `s < t` "*s is strictly less than t*" to mean `s ≤ t ∧ s ≠ t`.
* A preorder (on a set S) that is also antisymmetric is called a partial order

  on S.
* A partial order ≤ is called a total order if it also has the property that,

  for each s and t in S, either `s ≤ t` or `t ≤ s`.
* Suppose that ≤ is a partial order on a set S and s and t are elements of S. An element `j ∈ S` is said to be a join (or least upper bound) of s and t if `s ≤ j` and `t ≤ j`, AND for any element `k ∈ S` with `s ≤ k` and `t ≤ k`, we have `j ≤ k`.
* an element `m ∈ S` is a meet (or greatest lower bound) of s and t if `m ≤ s` and `m ≤ t`, AND for any element `n ∈ S` with `n ≤ s` and `n ≤ t`, we have `n ≤ m`
* a reflexive, transitive, and symmetric relation on a set S is called

  an equivalence on S.
* let R is a binary relation on a set S. The reflexive closure of R is the smallest reflexive relation R' that contains R. "Smallest" in the sense that if R'' is some other reflexive relation that contains all the pairs in R, then we have `R' ⊆ R''`
* the transitive closure of R is the smallest transitive relation R' that contains R
* The transitive closure of R is often written R+
* The reflexive and transitive closure of R is the smallest reflexive and transitive relation that contains R, often written R∗.
* Let preorder ≤ on a set S. A decreasing chain in ≤ is a sequence s1,s2,s3,... of elements of S such that each member of the sequence is strictly less than its predecessor: `si+1 < si` for every i.
* Chains can be either finite or infinite
* Suppose we have a set S with a preorder ≤. We say that ≤ is well

  founded if it contains no infinite decreasing chains. For example, the usual order on the natural numbers, with `0 < 1 < ...`, is well founded, but the same order on the integers, `...< −1 < 0 < 1 <...` is not. We sometimes omit mentioning ≤ explicitly and simply speak of S as a well-founded set.
