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Vidaara.orgClass 12 · Mathematics
CodeVID-M12-01-REL-01
Relations & Types of Relations — Assignment
Chapter: Relations and Functions
Topic: Relations & Types of Relations
Maximum Marks: 35
Time: 75 minutes
Name: ____________________ Roll No.: __________ Date: ____________

General Instructions

  • All questions are compulsory.
  • Section A carries 1 mark each, Section B 2 marks, Section C 3 marks and Section D 5 marks.
  • Show all working for Sections B, C and D. Only final answers are given at the end — for full solutions, raise your doubts with your teacher.
Section A — Multiple Choice Questions 5 × 1 = 5 marks
1.
On $A=\{1,2,3\}$, the relation $R=A\times A$ is called the:
  • A.empty relation
  • B.universal relation
  • C.identity relation
  • D.void relation
2.
The number of relations that can be defined on a set with $4$ elements is:
  • A.$16$
  • B.$256$
  • C.$2^{16}$
  • D.$4^{4}$
3.
A relation that is reflexive, symmetric and transitive is called:
  • A.a partial order
  • B.an equivalence relation
  • C.a function
  • D.the empty relation
4.
On $\mathbb{Z}$, the relation $a\,R\,b \iff a\le b$ is:
  • A.reflexive and symmetric
  • B.reflexive and transitive but not symmetric
  • C.an equivalence relation
  • D.symmetric and transitive only
5.
If $R$ on $\mathbb{Z}$ is $a\,R\,b \iff 5\mid(a-b)$, the number of distinct equivalence classes is:
  • A.$2$
  • B.$4$
  • C.$5$
  • D.infinitely many
Section B — Short Answer (2 marks) 4 × 2 = 8 marks
6.
Define a reflexive relation and give one example on the set $\{1,2,3\}$.
7.
Let $R=\{(1,2),(2,1),(1,1),(2,2)\}$ on $\{1,2\}$. State, with reason, whether $R$ is symmetric.
8.
Give an example of a relation on $\{1,2,3\}$ that is symmetric but not reflexive.
9.
On $\mathbb{Z}$, $a\,R\,b \iff a-b$ is even. Write the equivalence class $[1]$.
Section C — Short Answer (3 marks) 4 × 3 = 12 marks
10.
Check whether the relation $R$ on $\mathbb{R}$ given by $a\,R\,b \iff a\le b$ is reflexive, symmetric and transitive.
11.
On $A=\{1,2,3,4\}$, let $R=\{(a,b):a+b\text{ is even}\}$. Determine the type of relation.
12.
Show that the relation "is parallel to" on the set of all straight lines in a plane is symmetric and transitive.
13.
On $\mathbb{Z}$, $a\,R\,b \iff 3\mid(a-b)$. Find all equivalence classes.
Section D — Long Answer (5 marks) 2 × 5 = 10 marks
14.
Prove that the relation $R$ on $\mathbb{Z}$ defined by $a\,R\,b \iff (a-b)$ is divisible by $5$ is an equivalence relation. Hence write its equivalence classes.
15.
Let $A=\{1,2,3,\dots,9\}$ and $R$ on $A\times A$ be defined by $(a,b)\,R\,(c,d) \iff a+d=b+c$. Prove that $R$ is an equivalence relation.

Answer Key

Section A — Multiple Choice Questions
  1. (B) universal relation
  2. (C) $2^{16}$
  3. (B) an equivalence relation
  4. (B) reflexive and transitive but not symmetric
  5. (C) $5$
Section B — Short Answer (2 marks)
  1. A relation $R$ on $A$ is reflexive if $(a,a)\in R$ for every $a\in A$. Example: $R=\{(1,1),(2,2),(3,3)\}$.
  2. Yes, $R$ is symmetric.
  3. e.g. $R=\{(1,2),(2,1)\}$.
  4. $[1]=\{\dots,-3,-1,1,3,5,\dots\}$ (all odd integers).
Section C — Short Answer (3 marks)
  1. Reflexive and transitive, but not symmetric (hence not an equivalence relation).
  2. Equivalence relation; classes $\{1,3\}$ and $\{2,4\}$.
  3. Symmetric and transitive (in fact an equivalence relation).
  4. Three classes: $[0],[1],[2]$ (by remainder on division by $3$).
Section D — Long Answer (5 marks)
  1. Equivalence relation; equivalence classes $[0],[1],[2],[3],[4]$.
  2. $R$ is an equivalence relation.
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