Difference between revisions of "Relation"

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Revision as of 18:40, 1 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 $xRy$

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

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