CLASS XI | RELATION

Relation — Complete Chapter

Ordered pairs, Cartesian products, relations (domain, range, image, preimage), types (reflexive, symmetric, transitive, equivalence), with step‑by‑step examples and interactive diagrams.

Click any topic bar to expand/collapse. All math renders via MathJax; just paste this block into Blogger (HTML view).

Ordered Pair

Definition, equality/inequality, use in 2D & 3D

Definition of ordered pair

An ordered pair \((a,b)\) is a pair where the first component is \(a\) and the second is \(b\). Order matters: \((a,b)\neq(b,a)\) unless \(a=b\).

Example: Compare \((2,5)\) and \((5,2)\).

1) First components: \(2\) vs \(5\) differ. 2) Hence \((2,5)\neq(5,2)\).

Equality of ordered pair

\((a,b)=(c,d)\) iff \(a=c\) and \(b=d\).

Example: Solve \((x+1,\,3y)=(4,\,12)\).

1) Match first: \(x+1=4\Rightarrow x=3\).
2) Match second: \(3y=12\Rightarrow y=4\).

Inequality of ordered pair

\((a,b)\ne(c,d)\) if either \(a\ne c\) or \(b\ne d\).

Example: Show \((\sqrt{2},\,\pi)\ne(\sqrt{2},\,3.14)\) since \(\pi\ne 3.14\).

Use of ordered pair in 2D & 3D

Coordinates in 2D: \((x,y)\in\mathbb{R}^2\). In 3D: \((x,y,z)\in\mathbb{R}^3\). These identify unique points.

Example: Midpoint of \((x1,y1)\) and \((x2,y2)\) is \(\left(\frac{x1+x2}{2},\,\frac{y1+y2}{2}\right)\).

Cartesian Product

Definition, representation, forms, 2D/3D, properties

Definition of Cartesian product

For sets \(A,B\), the Cartesian product is \(A\times B=\{(a,b)\mid a\in A,\ b\in B\}\).

Example: If \(A=\{1,2\}, B=\{x,y\}\), then \(A\times B=\{(1,x),(1,y),(2,x),(2,y)\}\).

Representation of Cartesian product

Roster form lists all ordered pairs; set‑builder form uses a property.

Set‑builder & roster form

Example: \(A=\{0,1\}, B=\{2,3\}\).
Roster: \(A\times B=\{(0,2),(0,3),(1,2),(1,3)\}\).
Set‑builder: \(A\times B=\{(a,b)\mid a\in\{0,1\},\ b\in\{2,3\}\}\).

Formation of 2D & 3D using Cartesian product

\(\mathbb{R}^2=\mathbb{R}\times \mathbb{R}\), \(\mathbb{R}^3=\mathbb{R}\times \mathbb{R}\times \mathbb{R}\).

Example: A grid of integer points: \(\mathbb{Z}\times\mathbb{Z}=\{(m,n)\mid m,n\in\mathbb{Z}\}\).

Properties of Cartesian product

• \(A\times(B\cup C)=(A\times B)\cup(A\times C)\).
• \(A\times(B\cap C)=(A\times B)\cap(A\times C)\).
• \((A\times B)\times C\ne A\times(B\times C)\) (different grouping of pairs), but \(|A\times B|=|A|\cdot|B|\).

Example: If \(|A|=3, |B|=4\Rightarrow |A\times B|=12\).

Relation

Intro, definition, representation, domain/range/codomain, image/preimage, arrow diagram

Introduction of relation

A relation \(R\) from \(A\) to \(B\) is any subset of \(A\times B\). If \((a,b)\in R\), we write \(a\,R\,b\).

Definition of relation

Formally, \(R\subseteq A\times B\). If \(A=B\), it is a relation on \(A\).

Representation of relation

• Roster: list the pairs in \(R\).
• Set‑builder: \(R=\{(a,b)\in A\times B\mid \text{condition}\}\).
• Arrow diagram: draw elements of \(A\) and \(B\) with arrows from \(a\in A\) to \(b\in B\) if \((a,b)\in R\).

Roster form & set‑builder form of relation

Example: Let \(A=\{1,2,3\}, B=\{a,b\}\), relation “odd maps to \(a\), even maps to \(b\)”:
Roster: \(R=\{(1,a),(3,a),(2,b)\}\).
Set‑builder: \(R=\{(x,y)\in A\times B\mid (x\text{ odd}\Rightarrow y=a)\ \wedge\ (x\text{ even}\Rightarrow y=b)\}\).

Arrow diagram (interactive)

Sets

A = {1,2,3}, B = {a,b,c}

Toggle pairs (a,b) to include in relation R.

Live arrow diagram for R ⊆ A×B

Domain / Range / Codomain

Domain of relation

\(\mathrm{Dom}(R)=\{a\in A\mid \exists b\in B,\ (a,b)\in R\}\).

Range of relation

\(\mathrm{Ran}(R)=\{b\in B\mid \exists a\in A,\ (a,b)\in R\}\).

Codomain of relation

Codomain is the target set \(B\) (fixed with the relation’s definition), possibly larger than the range.

Image & preimage

\(\mathrm{Im}_R(a)=\{b\in B\mid (a,b)\in R\}\). For \(Y\subseteq B\), the preimage \(R^{-1}(Y)=\{a\in A\mid \exists y\in Y,\ (a,y)\in R\}\).

Example: For \(A=\{1,2,3\}\), \(B=\{a,b,c\}\), \(R=\{(1,a),(2,b),(2,c)\}\):

1) \(\mathrm{Dom}(R)=\{1,2\}\). 2) \(\mathrm{Ran}(R)=\{a,b,c\}\). 3) \(\mathrm{Im}_R(2)=\{b,c\}\).
4) For \(Y=\{a,c\}\), \(R^{-1}(Y)=\{1,2\}\).

Types of Relation

Void, universal, identity, reflexive, symmetric, transitive, equivalence, class

Void relation

On \(A\), \(R=\emptyset\) (contains no pairs). Not reflexive unless \(A=\emptyset\).

Example: On \(A=\{1,2\}\), \(\emptyset\) is not reflexive (\((1,1)\notin R\)).

Universal relation

On \(A\), \(R=A\times A\). Always reflexive, symmetric, and transitive.

Example: On \(A=\{1,2\}\), \(R=\{(1,1),(1,2),(2,1),(2,2)\}\).

Identity relation

\(I_A=\{(a,a)\mid a\in A\}\). Reflexive and symmetric? (Yes, trivially), transitive (Yes) → equivalence.

Example: On \(A=\{x,y\}\), \(I_A=\{(x,x),(y,y)\}\).

Reflexive relation

\(\forall a\in A,\ (a,a)\in R\).

Example: \(R=\{(1,1),(2,2)\}\) on \(A=\{1,2\}\) is reflexive.

Symmetric relation

\(\forall a,b\in A,\ (a,b)\in R \Rightarrow (b,a)\in R\).

Example: \(R=\{(1,2),(2,1)\}\) symmetric on \(A=\{1,2\}\).

Transitive relation

\(\forall a,b,c\in A,\ (a,b)\in R,\ (b,c)\in R \Rightarrow (a,c)\in R\).

Example: \(R=\{(1,2),(2,3),(1,3)\}\) is transitive on \(A=\{1,2,3\}\).

Equivalence relation

Reflexive, symmetric, and transitive. Partitions \(A\) into disjoint equivalence classes.

Example: On \(\mathbb{Z}\), “congruence mod \(3\)” is an equivalence relation. Classes: \([0],[1],[2]\).

Equivalence class of an element

\([a]=\{x\in A\mid x\,R\,a\}\). Distinct classes are disjoint or equal; their union is \(A\).

Example: In mod \(3\) on \(\mathbb{Z}\), \([4]=[1]=\{\dots,-5,-2,1,4,7,\dots\}\).

Interactive property checker (on A = {1,2,3})

Toggle pairs in R ⊆ A×A

A = {1,2,3}

Properties

Reflexive: … Symmetric: … Transitive: …

Equivalence = all three true.

Equivalence classes (if equivalence)

—

Question Practice

Mixed exercises with step‑by‑step solutions

Ordered pair

Q1. If \((2x-1,\,y+3)=(5,\,9)\), find \(x,y\).

Match components: \(2x-1=5\Rightarrow x=3\). \(y+3=9\Rightarrow y=6\).

Cartesian product

Q2. Let \(A=\{1,2\}, B=\{a,b,c\}\). Find \(|A\times B|\). List its elements.

\(|A\times B|=|A|\cdot|B|=2\cdot 3=6\).
Elements: \((1,a),(1,b),(1,c),(2,a),(2,b),(2,c)\).

Relation: domain/range

Q3. For \(R=\{(1,a),(2,a),(2,c)\}\) from \(A=\{1,2,3\}\) to \(B=\{a,b,c\}\), find domain, range, codomain.

\(\mathrm{Dom}(R)=\{1,2\}\), \(\mathrm{Ran}(R)=\{a,c\}\), \(\mathrm{Cod}(R)=B=\{a,b,c\}\).

Types of relation

Q4. On \(A=\{1,2,3\}\), let \(R=\{(1,1),(2,2),(3,3),(1,2),(2,1)\}\). Test for reflexive, symmetric, transitive; is \(R\) an equivalence?

Reflexive: yes (all \((a,a)\) present). Symmetric: yes (since \((1,2)\Rightarrow(2,1)\)).
Transitive: check \((1,2)\)&\((2,1)\Rightarrow(1,1)\) (in \(R\)), others okay ⇒ transitive.
Hence equivalence relation.