Set Theory/Relations

Ordered pairs

To define relations on sets we must have a concept of an ordered pair, as opposed to the unordered pairs the axiom of pair gives. To have a rigorous definition of ordered pair, we aim to satisfy one important property, namely, for sets a,b,c and d, $(a,b)=(c,d)⟺a=c\wedge b=d$.

As it stands, there are many ways to define an ordered pair to satisfy this property. A simple definition, then is $(a,b)=\{\{a\},\{a,b\}\}$. (This is true simply by definition. It is a convention that we can usefully build upon, and has no deeper significance.)

Relations

Using the definiton of ordered pairs, we now introduce the notion of a binary relation.

The simplest definition of a binary relation is a set of ordered pairs. More formally, a set $R$ is a relation if $z\in R\to z=(x,y)$ for some x,y. We can simplify the notation and write $(x,y)\in R$ or simply $xRy$.

We give a few useful definitions of sets used when speaking of relations.

• The domain of a relation R is defined as $\text{dom}R=\{x\mid \mathrm{\exists }y,(x,y)\in R\}$, or all sets that are the initial member of an ordered pair contained in R.

• The range of a relation R is defined as $\text{ran}R=\{y\mid \mathrm{\exists }x,(x,y)\in R\}$, or all sets that are the final member of an ordered pair contained in R.

• The union of the domain and range, $\text{field}R=\text{dom}R\cup \text{ran}R$, is called the field of R.

• A relation R is a relation on a set X if $\text{field}R\subseteq X$.

• The inverse of R is the set $R^{-1}=\{(y,x)\mid (x,y)\in R\}$

• The image of a set A under a relation R is defined as $R[A]=\{y\in \text{ran}R\mid \mathrm{\exists }x\in A,(x,y)\in R\}$.

• The preimage of a set B under a relation R is the image of B over R^-1 or $R^{-1}[B]=\{x\in \text{dom}R\mid \mathrm{\exists }y\in B,(x,y)\in R\}$

It is intuitive, when considering a relation to seek to construct more relations from it, or to combine it with others.

We can compose two relations, R and S to form one relation $S\circ R=\{(x,z)\mid \mathrm{\exists }y,(x,y)\in R\wedge (y,z)\in S\}$. So $xR\circ Sz$ means that there is some y such that $xRy\wedge ySz$.

We can define a few useful binary relations as examples:

1. The Cartesian Product of two sets is $A\times B=\{(a,b)\in 𝒫𝒫(A\cup B)\mid a\in A\wedge b\in B\}$, or the set where all elements of A are related to all elements of B. As an excercise, show that all relations from A to B are subsets of $A\times B$. Notationally $A\times A$ is written $A^2$
2. The membership relation on a set A, ${\in }_A=\{(a,b)\mid a,b\in A,a\in b\}$
3. The identity relation on A, $I_A=\{(a,b)\mid a,b\in A,a=b\}$

The following properties may or may not hold for a relation R on a set X:

• R is reflexive if $xRx$ holds for all x in X.
• R is symmetric if $xRy$ implies $yRx$ for all x and y in X.
• R is antisymmetric if $xRy$ and $yRx$ together imply that $x=y$ for all x and y in X.
• R is transitive if $xRy$ and $yRz$ together imply that $xRz$ holds for all x, y, and z in X.
• R is total if $xRy$, $yRx$, or both hold for all x and y in X.

Functions

Definitions

A function may be defined as a particular type of relation. We define a partial function $f:X\to Y$ as some mapping from a set $X$ to another set $Y$ that assigns to each $x\in X$ no more than one $y\in Y$. If on each $x\in X$, $f$ assigns exactly one $y\in Y$, then $f$ is called total function or just function. The following definitions are commonly used when discussing functions.

• If $f\subseteq X\times Y$ and $f$ is a function, then we can denote this by writing $f:X\to Y$. The set $X$ is known as the domain and the set $Y$ is known as the codomain.
• For a function $f:X\to Y$, the image of an element $x\in X$ is $y\in Y$ such that $f(x)=y$. Alternatively, we can say that $y$ is the value of $f$ evaluated at $x$.
• For a function $f:X\to Y$, the image of a subset $A$ of $X$ is the set $\{y\in Y:f(x)=y\text{ for some }x\in A\}$. This set is denoted by $f(A)$. Be careful not to confuse this with $f(x)$ for $x\in X$, which is an element of $Y$.
• The range of a function $f:X\to Y$ is $f(X)$, or all of the values $y\in Y$ where we can find an $x\in X$ such that $f(x)=y$.
• For a function $f:X\to Y$, the preimage of a subset $B$ of $Y$ is the set $\{x\in X:f(x)\in B\}$. This is denoted by $f^{-1}(B)$.

Properties of functions

A function $f:X\to Y$ is onto, or surjective, if for each $y\in Y$ exists $x\in X$ such that $f(x)=y$. It is easy to show that a function is surjective if and only if its codomain is equal to its range. It is one-to-one, or injective, if different elements of $X$ are mapped to different elements of $Y$, that is $f(x)=f(y)\Rightarrow x=y$. A function that is both injective and surjective is intuitively termed bijective.

Composition of functions

Given two functions $f:X\to Y$ and $g:Y\to Z$, we may be interested in first evaluating f at some $x\in X$ and then evaluating g at $f(x)$. To this end, we define the composition' of these functions, written $g\circ f$, as

$(g\circ f)(x)=g(f(x))$

Note that the composition of these functions maps an element in $X$ to an element in $Z$, so we would write $g\circ f:X\to Z$.

Inverses of functions

If there exists a function $g:Y\to X$ such that for $f:X\to Y$, $g\circ f=I_X$, we call $g$ a left inverse of $f$. If a left inverse for $f$ exists, we say that $f$ is left invertible. Similarly, if there exists a function $h:Y\to X$ such that $f\circ h=I_Y$ then we call $h$ a right inverse of $f$. If such an $h$ exists, we say that $f$ is right invertible. If there exists an element which is both a left and right inverse of $f$, we say that such an element is the inverse of $f$ and denote it by $f^{-1}$. Be careful not to confuse this with the preimage of f; the preimage of f always exists while the inverse may not. Proof of the following theorems is left as an exercise to the reader.

Theorem: If a function has both a left inverse $g$ and a right inverse $h$, then $g=h=f^{-1}$.

Theorem: A function is invertible if and only if it is bijective.