JEE Main Sets and Relations Practice Questions With Solutions

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JEE Main Mathematics Sets and Relations Practice Questions

Question 1.

Let $A = {2, 3, 6, 8, 9, 11}$ and $B = {1, 4, 5, 10, 15}$ . Let $R$ be a relation on $A \times B$ defined by $(a, b) R (c, d)$ if and only if $3 a d - 7 b c$ is an even integer. Then the relation $R$ is

reflexive but not symmetric.

an equivalence relation.

reflexive and symmetric but not transitive.

transitive but not symmetric.

Question 2.

Let $A = {1, 2, 3, 4, 5}$ . Let $R$ be a relation on $A$ defined by $x R y$ if and only if $4 x \leq 5 y$ . Let $m$ be the number of elements in $R$ and $n$ be the minimum number of elements from $A \times A$ that are required to be added to R to make it a symmetric relation. Then m + n is equal to :

Question 3.

Let $A = {n \in [100, 700] \cap N : n$ is neither a multiple of 3 nor a multiple of 4 $}$ . Then the number of elements in $A$ is

300

310

290

280

Question 4.

Let the relations $R_{1}$ and $R_{2}$ on the set $X = {1, 2, 3, \dots, 20}$ be given by $R_{1} = {(x, y) : 2 x - 3 y = 2}$ and $R_{2} = {(x, y) : - 5 x + 4 y = 0}$ . If $M$ and $N$ be the minimum number of elements required to be added in $R_{1}$ and $R_{2}$ , respectively, in order to make the relations symmetric, then $M + N$ equals

Question 5.

Let a relation $R$ on $N \times N$ be defined as: $(x_{1}, y_{1}) R (x_{2}, y_{2})$ if and only if $x_{1} \leq x_{2}$ or $y_{1} \leq y_{2}$ .Consider the two statements:

(I) $R$ is reflexive but not symmetric.

(II) $R$ is transitive

Then which one of the following is true?

Only (II) is correct.

Both (I) and (II) are correct.

Neither (I) nor (II) is correct.

Only (I) is correct.

Question 6.

Consider the relations

R_{1}

and

R_{2}

defined as

a R_{1} b \Leftrightarrow a^{2} + b^{2} = 1

for all

a, b \in R

and

(a, b) R_{2} (c, d) \Leftrightarrow

a + d = b + c

for all

(a, b), (c, d) \in N \times N

. Then :

R_{1}

and

R_{2}

both are equivalence relations

Only

R_{1}

is an equivalence relation

Only

R_{2}

is an equivalence relation

Neither

R_{1}

nor

R_{2}

is an equivalence relation

Question 7.

If R is the smallest equivalence relation on the set ${1, 2, 3, 4}$ such that ${(1, 2), (1, 3)} \subset R$ , then the number of elements in $R$ is __________.

Question 8.

Let $R$ be a relation on $Z \times Z$ defined by $(a, b) R (c, d)$ if and only if $a d - b c$ is divisible by 5. Then $R$ is

Reflexive and transitive but not symmetric

Reflexive and symmetric but not transitive

Reflexive but neither symmetric nor transitive

Reflexive, symmetric and transitive

Question 9.

Let $A$ and $B$ be two finite sets with $m$ and $n$ elements respectively. The total number of subsets of the set $A$ is 56 more than the total number of subsets of $B$ . Then the distance of the point $P (m, n)$ from the point $Q (- 2, - 3)$ is :

Question 10.

Let

S = {1, 2, 3, \dots, 10}

. Suppose

M

is the set of all the subsets of

S

, then the relation

R = {(A, B) : A \cap B \neq ϕ; A, B \in M}

is :

symmetric only

reflexive only

symmetric and reflexive only

symmetric and transitive only

Question 1.

Let $A = {1, 3, 4, 6, 9}$ and $B = {2, 4, 5, 8, 10}$ . Let $R$ be a relation defined on $A \times B$ such that $R = {((a_{1}, b_{1}), (a_{2}, b_{2})) : a_{1} \leq b_{2}$ and $b_{1} \leq a_{2}}$ . Then the number of elements in the set R is :

180

160

Question 2.

An organization awarded 48 medals in event 'A', 25 in event 'B' and 18 in event 'C'. If these medals went to total 60 men and only five men got medals in all the three events, then, how many received medals in exactly two of three events?

Question 3.

Let $A = {2, 3, 4}$ and $B = {8, 9, 12}$ . Then the number of elements in the relation $R = {((a_{1}, b_{1}), (a_{2}, b_{2})) \in (A \times B, A \times B) : a_{1}$ divides $b_{2}$ and $a_{2}$ divides $b_{1}}$ is :

Question 4.

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

reflexive but neither symmetric nor transitive

transitive but neither symmetric nor reflexive

symmetric but neither reflexive nor transitive

an equivalence relation

Question 5.

Let $P (S)$ denote the power set of $S = {1, 2, 3, \dots ., 10}$ . Define the relations $R_{1}$ and $R_{2}$ on $P (S)$ as ${AR}_{1} B$ if $(A \cap B^{c}) \cup (B \cap A^{c}) = \emptyset$ and ${AR}_{2} B$ if $A \cup B^{c} = B \cup A^{c}, \forall A, B \in P (S)$ . Then :

only

R_{2}

is an equivalence relation

both

R_{1}

and

R_{2}

are not equivalence relations

both

R_{1}

and

R_{2}

are equivalence relations

only

R_{1}

is an equivalence relation

Question 6.

Let $R$ be a relation on $R$ , given by $R = {(a, b) : 3 a - 3 b + \sqrt{7}$ is an irrational number $}$ . Then $R$ is

an equivalence relation

reflexive and symmetric but not transitive

reflexive and transitive but not symmetric

reflexive but neither symmetric nor transitive

Question 7.

Among the relations

S = {(a, b) : a, b \in R - {0}, 2 + \frac{a}{b} > 0}

and

T = {(a, b) : a, b \in R, a^{2} - b^{2} \in Z}

S

is transitive but

T

is not

both

S

and

T

are symmetric

neither

S

nor

T

is transitive

T

is symmetric but

S

is not

Question 8.

Let $R$ be a relation on $N \times N$ defined by $(a, b) R (c, d)$ if and only if $a d (b - c) = b c (a - d)$ . Then $R$ is

symmetric and transitive but not reflexive

reflexive and symmetric but not transitive

transitive but neither reflexive nor symmetric

symmetric but neither reflexive nor transitive

Question 9.

The minimum number of elements that must be added to the relation $R = {(a, b), (b, c)}$ on the set ${a, b, c}$ so that it becomes symmetric and transitive is :

Question 10.

Let R be a relation defined on $N$ as $a R b$ if $2 a + 3 b$ is a multiple of $5, a, b \in N$ . Then R is

an equivalence relation

non reflexive

symmetric but not transitive

transitive but not symmetric

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JEE Main Sets and Relations Practice Questions With Solutions

JEE Main Mathematics Sets and Relations Practice Questions

Question 1.

Question 2.

Question 3.

Question 4.

Question 5.

Question 6.

Question 7.

Question 8.

Question 9.

Question 10.

Question 1.

Question 2.

Question 3.

Question 4.

Question 5.

Question 6.

Question 7.

Question 8.

Question 9.

Question 10.

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