Let A = Z , the set of integers. Let R 1 = { ( m , n ) ∈ Z × Z : ( m + 4 n ) is divisible by 5 in Z } , Let R 2 = { ( m , n ) ∈ Z × Z : ( m + 9 n ) is divisibile by 5 in Z Which one of the following is correct?

R 1 is not a symmetric relation on Z

Consider the following relations: R : { ( x , y ) ∣ x , y are real numbers and x = w y for some rational number w } S = { m n , p q ∣ m , n , p and q are integers such that n , q ≠ 0 and q m = p n } Then

S is an equivalence relation but R is not an equivalence relation

neither R nor S is an equivalence relation

R and S both are equivalence relations

R is an equivalence relation but S is not an equivalence relation

E M B I B E

Alpha Question Bank for Engineering: Mathematics>Chapter 2 - Set Theory and Relations>Exercise-2>Q 2

EASY

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IMPORTANT

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Let $A = N \times N$ be the Cartesian product of $N$ and $N$ . Let $S = \{((m, n), (p, q)) \in A \times A : m + q = n + p\}$ .

Consider the following statements

$I .$ If $((m, n), (p, q)) \in S,$ and $((p, q), (r, s)) \in S$ then $((r, s), (m, n)) \in S$
$I I .$ There exists at least one element $((m, n), (p, q)) \in S$ such that $((p, q), (m, n)) \notin S$
Which of the statements given above is/are correct?

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Important Questions on Set Theory and Relations

HARD

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IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 2 - Set Theory and Relations>Exercise-2>Q 3

Let

A = Z,

the set of integers. Let

R_{1} = {(m, n) \in Z \times Z : (m + 4 n)

is divisible by

5

in

Z}

, Let

R_{2} = {(m, n) \in Z \times Z : (m + 9 n)

is divisibile by

5

in

Z

Which one of the following is correct?

HARD

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IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 2 - Set Theory and Relations>Exercise-2>Q 4

Let

X

be the set of all persons living in a state. Elements

x, y

in

X

are said to be related if

{}^{'}x < y^{'}

whenever

y

is 5 years older than

x .

Which one of the following is correct?

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IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 2 - Set Theory and Relations>Exercise-2>Q 5

Let $A = \{1, 2, 3, 4\}$ and $R$ be a relation in $A$ given, then relation $R$ is by $R = \{(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 1), (3, 1), (1, 3)\}$

HARD

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IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 2 - Set Theory and Relations>Exercise-2>Q 6

For

n, m \in N, n ∣ m

means that

n

is a factor of

m,

then relation

∣

is

MEDIUM

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Alpha Question Bank for Engineering: Mathematics>Chapter 2 - Set Theory and Relations>Exercise-3>Q 1

Let

P = \{θ : \sin θ - \cos θ = \sqrt{2} \cos θ\}

and

Q = \{θ : \sin θ + \cos θ = \sqrt{2} \sin θ\}

be two sets. Then

HARD

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IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 2 - Set Theory and Relations>Exercise-3>Q 2

Consider the following relations:
$R : {(x, y) ∣ x, y$ are real numbers and $x = w y$ for some rational number $w}$

$S = {(\frac{m}{n}, \frac{p}{q}) ∣ m, n, p$ and $q$ are integers such that $n, q \neq 0$ and $q m = p n}$
Then

HARD

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IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 2 - Set Theory and Relations>Exercise-3>Q 3

Let $R$ be the set of real numbers.

Statement- $1 : A = {(x, y) \in R \times R : y - x$ is an integer} is an equivalence relation on $R .$

Statement- $2 : B = {(x, y) \in R \times R : x = α y$ for some rational number $α)$ is an equivalence relation on $R .$

HARD

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IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 2 - Set Theory and Relations>Exercise-3>Q 4

Consider the following relation $R$ , on the set of real square matrices of order $3$ .

$R = {(A, B) ∣ A = P^{- 1} B P$ , for some invertible matrix $P}$ .

Statement - $1 : R$ , is equivalence relation.

Statement - $2 :$ For any two invertible $3 \times 3$ matrices $M$ and $N, {(MN)}^{- 1} = N^{- 1} M^{- 1} .$

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Let A=N×N be the Cartesian product of N and N. Let S=m,n,p,q∈A×A:m+q=n+p. Consider the following statements I. If m,n,p,q∈S, and p,q,r,s∈S then r,s,m,n∈S II. There exists at least one element m,n,p,q∈S such that p,q,m,n∉S Which of the statements given above is/are correct?

Important Questions on Set Theory and Relations

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