Difference between revisions of "Relation"

Revision as of 22:12, 1 March 2015 (view source) Alec (Talk | contribs) m ← Older edit		Revision as of 22:21, 4 March 2015 (view source) Alec (Talk | contribs) m Newer edit →
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	Rather than writing {{M|(x,y)\in R}} to say {{M|x}} and {{M|y}} are related we can instead say {{M|xRy}}		Rather than writing {{M|(x,y)\in R}} to say {{M|x}} and {{M|y}} are related we can instead say {{M|xRy}}
	==Basic terms==		==Basic terms==
		+	Proof that domain, range and field exist may be found [[The domain, range and field of a relation exist|here]]
	===Domain===		===Domain===
	The set of all {{M|x}} which are related by {{M|R}} to some {{M|y}} is the domain.		The set of all {{M|x}} which are related by {{M|R}} to some {{M|y}} is the domain.

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Revision as of 22:21, 4 March 2015

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

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

$$R^{-1}$$