Difference between revisions of "Relation"

Revision as of 22:07, 1 March 2015 (view source) Alec (Talk | contribs) m ← Older edit		Revision as of 22:12, 1 March 2015 (view source) Alec (Talk | contribs) m Newer edit →
Line 4:		Line 4:
	==Notation==		==Notation==
	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==
−	==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.
	<math>\text{Dom}(R)=\{x|\exists\ y: xRy\}</math>		<math>\text{Dom}(R)=\{x|\exists\ y: xRy\}</math>
−	==Range==	+	===Range===
	The set of all {{M|y}} which are a relation of some {{M|x}} by {{M|R}} is the range.		The set of all {{M|y}} which are a relation of some {{M|x}} by {{M|R}} is the range.
	<math>\text{Ran}(R)=\{y|\exists\ x: xRy\}</math>		<math>\text{Ran}(R)=\{y|\exists\ x: xRy\}</math>
−	==Field==	+	===Field===
	The set <math>\text{Dom}(R)\cup\text{Ran}(R)=\text{Field}(R)</math>		The set <math>\text{Dom}(R)\cup\text{Ran}(R)=\text{Field}(R)</math>
−	==Relation in X==	+	===Relation in X===
	To be a relation in a set {{M|X}} we must have <math>\text{Field}(R)\subset X</math>		To be a relation in a set {{M|X}} we must have <math>\text{Field}(R)\subset X</math>
−	==Image of A under R==	+	==Images of sets==
		+	===Image of A under R===
	This is just the set of things that are related to things in A, denoted <math>R[A]</math>		This is just the set of things that are related to things in A, denoted <math>R[A]</math>
	<math>R[A]=\{y\in\text{Ran}(R)|\exists x\in A:xRa\}</math>		<math>R[A]=\{y\in\text{Ran}(R)|\exists x\in A:xRa\}</math>
−	==Inverse image of B under R==	+	===Inverse image of B under R===
	As you'd expect this is the things that are related to things in B, denoted <math>R^{-1}[B]</math>		As you'd expect this is the things that are related to things in B, denoted <math>R^{-1}[B]</math>
	<math>R^{-1}[B]=\{x\in\text{Dom}(R)|\exists y\in B:xRy\}</math>		<math>R^{-1}[B]=\{x\in\text{Dom}(R)|\exists y\in B:xRy\}</math>
		+
		+	===Important lemma===
		+	It is very important to know that the inverse image of B under R is the same as the image under <math>R^{-1}</math>
	{{Definition|Set Theory}}		{{Definition|Set Theory}}

---

Revision as of 22:12, 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 $xRy$

Basic terms

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

$$R^{-1}$$