Determine the relation R on the set of whole numbers ≤ 10 defined by R = { ( x , y ) ∣ 2 x + 3 y = 12 } .

Given, R = ( x , y ) ; 2 x + 3 y = 12 , { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 910 } x = 0 , 2 × 0 + 3 y = 12 ⇒ y = 4 ⇒ ( 0 , 4 ) x = 1 , 2 × 1 + 3 y = 12 ⇒ y = 10 3 x = 2 , 2 × 2 + 37 = 12 ⇒ y = 8 3 x = 3 , 2 × 3 + 3 y = 12 ⇒ y = 2 ⇒ ( 3 , 2 ) x = 4 , 2 × 4 + 3 y = 12 ⇒ y = 4 3 x = 5 , 2 × 5 + 3 y = 12 ⇒ y = 2 3 x = 6 , 2 × 6 + 3 y = 12 ⇒ y = 0 ⇒ ( 6 , 0 ) ∴ R = { ( 0 , 4 ) , ( 3 , 2 ) , ( 6 , 0 ) }

If A = { 2 , 4 } , B = { 3 , 4 , 5 } , then ( A ∩ B ) × ( A ∪ B ) =

{ ( 4 , 2 ) , ( 4 , 3 ) , ( 4 , 4 ) , ( 4 , 5 ) }

{ ( 3 , 2 ) , ( 3 , 4 ) , ( 4 , 4 ) , ( 5 , 4 ) }

{ ( 4 , 3 ) , ( 4 , 4 ) , ( 4 , 5 ) }

{ ( 2 , 3 ) , ( 2 , 4 ) , ( 2 , 5 ) }

Let N denote the set of all natural numbers. Define two binary relations on N as R 1 = x , y ∈ N × N : 2 x + y = 10 and R 2 = x , y ∈ N × N : x + 2 y = 10 . Then

both R 1 and R 2 are transitive relations

both R 1 and R 2 are symmetric relations

EASY

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Cartesian product of sets satisfies the commutative property.

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Important Questions on Sets

MEDIUM

Mathematics>Algebra>Sets>Cartesian Product of Two Sets

Let

A = \{2, 3, 4, 5, \dots ., 30\}

and

' ≃'

be an equivalence relation on

A \times A,

defined by

(a, b) ≃ (c, d),

if and only if

a d = b c

. Then the number of ordered pairs which satisfy this equivalence relation with ordered pair

(4, 3)

is equal to :

EASY

Mathematics>Algebra>Sets>Cartesian Product of Two Sets

Consider the following relations in the real numbers

R_{1} = \{(x, y) ∣ x^{2} + y^{2} \leq 25\}

and

R_{2} = \{(x, y), y \geq \frac{4 x^{2}}{9}\}

, then the range of

R_{1} \cap R_{2}

MEDIUM

Mathematics>Algebra>Sets>Cartesian Product of Two Sets

Let

A

be a set consisting of

10

elements. The number of non-empty relations from

A

A

that are reflexive but not symmetric is

MEDIUM

Mathematics>Algebra>Sets>Cartesian Product of Two Sets

Suppose

P, Q

and

R

are three sets, each with three elements. The number of subsets of the set

P \times Q \times R,

that have at least

2

elements is

MEDIUM

Mathematics>Algebra>Sets>Cartesian Product of Two Sets

n (A) = 2

and total number of possible relations from set

A

to set

B

1024

, then

n (B)

MEDIUM

Mathematics>Algebra>Sets>Cartesian Product of Two Sets

Suppose

A, B

and

C

are three sets, each with three elements. The number of subsets of the set

A \times B \times C

that have at least

2

elements is

MEDIUM

Mathematics>Algebra>Sets>Cartesian Product of Two Sets

If two sets

A

and

B

have

99

elements in common, then the number of elements common to the sets

A \times B

and

B \times A

EASY

Mathematics>Algebra>Sets>Cartesian Product of Two Sets

Determine the relation $R$ on the set of whole numbers $\leq 10$ defined by $R = {(x, y) ∣ 2 x + 3 y = 12}$ .

EASY

Mathematics>Algebra>Sets>Cartesian Product of Two Sets

A = {2, 4}, B = {3, 4, 5},

then

(A \cap B) \times (A \cup B) =

HARD

Mathematics>Algebra>Sets>Cartesian Product of Two Sets

The number of ordered pairs

(a, b)

of positive integers such that

\frac{2 a - 1}{b}

and

\frac{2 b - 1}{a}

are both integers is

EASY

Mathematics>Algebra>Sets>Cartesian Product of Two Sets

A

and

B

are two non-empty finite sets having

3

and

4

elements respectively, then what would be the number of ordered pairs in

A \times B ?

EASY

Mathematics>Algebra>Sets>Cartesian Product of Two Sets

Let

A = {2, 3, 4, 5}, B = {36, 45, 49, 60, 77, 90}

and let

R

be the relation 'is factor of' from

A

B

. Then the range of

R

is the set

EASY

Mathematics>Algebra>Sets>Cartesian Product of Two Sets

If there are

2

elements in a set

A,

then what would be the number of possible relations from the set

A

to set

A ?

EASY

Mathematics>Algebra>Sets>Cartesian Product of Two Sets

Let

A, B, C

are three non-empty sets. The number of relations from

A

B

64

, that of

B

C

4096

and that of

A

C

256

. Then the numbers of elements of the sets are in

EASY

Mathematics>Algebra>Sets>Cartesian Product of Two Sets

A

and

B

are non-singleton sets and

n (A \times B) = 35

. If

B \subset A

, then

{}^{n (A)}C_{n (B)} =

EASY

Mathematics>Algebra>Sets>Cartesian Product of Two Sets

If set

A = {x : x is a measure of an angle of scalen triangle}

, then the number of relation containing (measure of minimum angle, measure of maximum angle) is

MEDIUM

Mathematics>Algebra>Sets>Cartesian Product of Two Sets

Let

A = {x : 1 < x < 10, x is an odd natural number}

and

B = {y : 90 < y < 100, y is a prime number}

. Write the number of relations from

A

B

EASY

Mathematics>Algebra>Sets>Cartesian Product of Two Sets

Let

Y = \{1, 2, 3, 4, 5\}, A = \{1, 2\}, B = \{3, 4, 5\}

and

ϕ

denotes null set. If

(A \times B)

denotes cartesian product of the sets

A

and

B

, then

(Y \times A) \cap (Y \times B)

MEDIUM

Mathematics>Algebra>Sets>Cartesian Product of Two Sets

R = \{(x, y) : x, y \in Z, x^{2} + 3 y^{2} \leq 8\}

is a relation on the set of integers

Z

, then the domain of

R^{- 1}

EASY

Mathematics>Algebra>Sets>Cartesian Product of Two Sets

Let

N

denote the set of all natural numbers. Define two binary relations on

N

R_{1} = \{(x, y) \in N \times N : 2 x + y = 10\}

and

R_{2} = \{(x, y) \in N \times N : x + 2 y = 10\}

. Then

Cartesian product of sets satisfies the commutative property.

Important Questions on Sets

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