Difference between revisions of "Relation"

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	A set {{M|R}} is a binary relation if all elements of {{M|R}} are [[Ordered pair|ordered pairs]]. That is for any {{M|z\in R\ \exists x\text{ and }y:(x,y)}}		A set {{M|R}} is a binary relation if all elements of {{M|R}} are [[Ordered pair|ordered pairs]]. That is for any {{M|z\in R\ \exists x\text{ and }y:(x,y)}}
		+	[[Function|Functions]], [[Equivalence relation|equivalence relations]] and [[Ordering|orderings]] are special kinds of relation
	==Notation==		==Notation==

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Revision as of 16:42, 5 March 2015

A set $R$ is a binary relation if all elements of $R$ are ordered pairs. That is for any $z\in R\ \exists x\text{ and }y:(x,y)$

Functions, equivalence relations and orderings are special kinds of relation

Notation

Rather than writing $(x,y)\in R$ to say $x$ and $y$ are related we can instead say $xRy$

Basic terms

Proof that domain, range and field exist may be found here

Domain

The set of all $x$ which are related by $R$ to some $y$ is the domain.

$$\text{Dom}(R)=\{x|\exists\ y: xRy\}$$

Range

The set of all $y$ which are a relation of some $x$ by $R$ is the range.

$$\text{Ran}(R)=\{y|\exists\ x: xRy\}$$

Field

The set

$$\text{Dom}(R)\cup\text{Ran}(R)=\text{Field}(R)$$

Relation in X

To be a relation in a set $X$ we must have

$$\text{Field}(R)\subset X$$

Images of sets

Image of A under R

This is just the set of things that are related to things in A, denoted

$$R[A]$$

$$R[A]=\{y\in\text{Ran}(R)|\exists x\in A:xRa\}$$

Inverse image of B under R

As you'd expect this is the things that are related to things in B, denoted

$$R^{-1}[B]$$

$$R^{-1}[B]=\{x\in\text{Dom}(R)|\exists y\in B:xRy\}$$

Important lemma

It is very important to know that the inverse image of B under R is the same as the image under

$$R^{-1}$$

Properties of relations

Symmetric

A relation $R$ in $A$ is symmetric if for all $a,b\in A$ we have that $aRb\implies bRa$ - a property of equivalence relations

Antisymmetric

A binary relation $R$ in $A$ is antisymmetric if for all $a,b\in A$ we have

$$aRb\text{ and }bRA\implies a=b$$
Symmetric implies elements are related to each other, antisymmetric implies only the same things are related to each other.

Reflexive

For a relation $R$ and for all $a\in A$ we have $aRa$ - $a$ is related to itself.

Transitive

A relation $R$ in $A$ is transitive if for all $a,b,c\in A$ we have

$$[aRb\text{ and }bRc\implies aRc]$$

Asymmetric

A relation $S$ in $A$ is asymmetric if $aSb\implies(b,a)\notin S$, for example $<$ has this property, we can have $a<b$ or $b<a$ but not both.