Relations; types of relations; equivalence relations — practice questions

Question 1 · difficulty L2 · understanding

The relation $R=\{(x, y): x, y \in \mathbb{Z}$ and $x+y$ is even $\}$ is:

• A. reflexive and transitive but not symmetric
• B. reflexive and symmetric but not transitive
• C. an equivalence relation
• D. symmetric and transitive but not reflexive

Question 2 · difficulty L2 · understanding

Let $$\mathrm{A}=\{1,2,3,4,5,6,7\}$$. Then the relation $$\mathrm{R}=\{(x, y) \in \mathrm{A} \times \mathrm{A}: x+y=7\}$$ is :

• A. reflexive but neither symmetric nor transitive
• B. transitive but neither symmetric nor reflexive
• C. symmetric but neither reflexive nor transitive
• D. an equivalence relation

Question 3 · difficulty L2 · understanding

Let $$R_{1}$$ and $$R_{2}$$ be two relations defined on $$\mathbb{R}$$ by $$a \,R_{1} \,b \Leftrightarrow a b \geq 0$$ and $$a \,R_{2} \,b \Leftrightarrow a \geq b$$ Then,

• A. $$R_{1}$$ is an equivalence relation but not $$R_{2}$$
• B. $$R_{2}$$ is an equivalence relation but not $$R_{1}$$
• C. both $$R_{1}$$ and $$R_{2}$$ are equivalence relations
• D. neither $$R_{1}$$ nor $$R_{2}$$ is an equivalence relation

Question 4 · difficulty L2 · understanding

Let N be the set of natural numbers and a relation R on N be defined by $$R = \{ (x,y) \in N \times N:{x^3} - 3{x^2}y - x{y^2} + 3{y^3} = 0\}$$. Then the relation R is :

• A. symmetric but neither reflexive nor transitive
• B. reflexive but neither symmetric nor transitive
• C. reflexive and symmetric, but not transitive
• D. an equivalence relation

Question 5 · difficulty L2 · understanding

Let R = {(P, Q) | P and Q are at the same distance from the origin} be a relation, then the equivalence class of (1, $$-$$1) is the set :

• A. $$S = \{ (x,y)|{x^2} + {y^2} = \sqrt 2 \}$$
• B. $$S = \{ (x,y)|{x^2} + {y^2} = 2\}$$
• C. $$S = \{ (x,y)|{x^2} + {y^2} = 1\}$$
• D. $$S = \{ (x,y)|{x^2} + {y^2} = 4\}$$

Question 6 · difficulty L2 · understanding

Let $R=\{(3,3),(6,6),(9,9),(12,12),(6,12)$, $(3,9),(3,12),(3,6)\}$ be a relation on the set $A=\{3,6,9,12\}$. The relation is :

• A. reflexive and symmetric only
• B. an equivalence relation
• C. reflexive only
• D. reflexive and transitive only

Question 7 · difficulty L2 · understanding

Let $R=\{(1,3),(4,2),(2,4),(2,3),(3,1)\}$ be a relation on the set $A=\{1,2,3,4\}$. The relation $R$ is :

• A. a function
• B. transitive
• C. not symmetric
• D. reflexive

Question 8 · difficulty L3 · understanding

Consider the relation $R$ on the set $\{-2,-1,0,1,2\}$ defined by $(a, b) \in R$ if and only if $1+a b>0$. Then, among the statements : I. The number of elements in R is 17 II. R is an equivalence relation

• A. Only I is true
• B. Only II is true
• C. Both I and II are true
• D. Neither I nor II is true

Question 9 · difficulty L3 · understanding

Let $\mathrm{A}=\{0,1,2, \ldots, 9\}$. Let R be a relation on A defined by $(x, y) \in \mathrm{R}$ if and only if $|x-y|$ is a multiple of 3. Given below are two statements : Statement I : $n(\mathrm{R})=36$. Statement II : R is an equivalence relation. In the light of the above statements, choose the correct answer from the options given below :

• A. Statement I is correct but Statement II is incorrect
• B. Both Statement I and Statement II are correct
• C. Both Statement I and Statement II are incorrect
• D. Statement I is incorrect but Statement II is correct

Question 10 · difficulty L3 · understanding

Let $\mathrm{A}=\{-2,-1,0,1,2,3,4\}$. Let R be a relation on A defined by $x \mathrm{R} y$ if and only if $2 x+y \leqslant 2$. Let $l$ be the number of elements in R . Let m and n be the minimum number of elements required to be added in R to make it reflexive and symmetric relations respectively. Then $\mathrm{l}+\mathrm{m}+\mathrm{n}$ is equal to :

• A. 34
• B. 32
• C. 33
• D. 35