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, $(a,b)=(c,d)\iff a=c\wedge b=d$ .

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.)

Theorem

$(a,b)=(c,d)\iff a=c\wedge b=d$

Proof

If $a=c$ and $b=d$ , then $(a,b)=\{\{a\},\{a,b\}\}=\{\{c\},\{c,d\}\}=(c,d)$ .
Now, if $(a,b)=(c,d)$ then $\{\{a\},\{a,b\}\}=\{\{c\},\{c,d\}\}$ . Then $\cap \{\{a\},\{a,b\}\}=\cap \{\{c\},\{c,d\}\}$ , so $\{a\}=\{c\}$ and a=c.
So we have $(a,b)=(a,d)$ . Thus $\cup \{\{a\},\{a,b\}\}=\cup \{\{a\},\{a,d\}\}$ meaning $\{a,b\}=\{a,d\}$ .

a=b

, we have

\{b\}=\{b,d\}

and thus

d\in \{b\}

b=d

a\neq b

, note

b\in \{a,d\}

, so

b=d

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 $z\in R\rightarrow z=(x,y)$ 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 ${\mbox{dom}}\ R=\{x\mid \exists y,(x,y)\in R\}$ , or all sets that are the initial member of an ordered pair contained in R.

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

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

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

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 $R[A]=\{y\in {\mbox{ran}}\ R\mid \exists x\in A,(x,y)\in R\}$ .

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

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 $S\circ R=\{(x,z)\mid \exists y,(x,y)\in R\wedge (y,z)\in S\}$ . So $xS\circ Rz$ means that there is some y such that $xRy\wedge ySz$ .

We can define a few useful binary relations as examples:

The Cartesian Product of two sets is $A\times B=\{(a,b)\in {\mathcal {P}}({\mathcal {P}}(A\cup B))\mid a\in A\wedge b\in B\}$ , or the set where all elements of A are related to all elements of B. As an exercise, show that all relations from A to B are subsets of $A\times B$ . Notationally $A\times A$ is written $A^{2}$
The membership relation on a set A, $\in _{A}=\{(a,b)\mid a,b\in A,a\in b\}$
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 $f:X\rightarrow Y$ as some mapping from a set $X$ to another set $Y$ that assigns to each $x\in X$ no more than one $y\in Y$ . Alternatively, f is a function if and only if $f\circ f^{-1}\subseteq I_{Y}$

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 $\{y\in Y:f(x)=y{\mbox{ for some }}x\in A\}$ . 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 $f:X\rightarrow Y$ 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 $f:X\rightarrow Y$ and $g:Y\rightarrow Z$ , 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 $g\circ f:X\rightarrow Z$ .

Inverses of functions

If there exists a function $g:Y\rightarrow X$ such that for $f:X\rightarrow Y$ , $g\circ f=I_{X}$ , 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 $h:Y\rightarrow X$ 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.

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