Relations Test 6

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Relations Test 6

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Weekly Quiz Competition

Question 1
1 / -0

Let A={ 1, 2, 3, 4} and R= {( 2, 2), (3, 3), (4, 4), (1, 2)} be a relation on A. Then R is

A
Reflexive

B
Symmetric

C
Transitive

D
None of these

Solution

$$\neq$$ R as every element is not related to itself.
$$\neq$$ S as (1, 2) $$\in$$ R but (2, 1) $$\notin$$ R
T (1,2), $$\in$$ R, (2, 2) $$\in$$ R
$$\Rightarrow $$ (1, 2)o (2, 2) = (1, 2) $$\in$$ R
Question 2
1 / -0

Given the relation R= {(1,2), (2,3) } on the set {1, 2, 3}, the minimum number of ordered pairs which when added to R make it an equivalence relation is

A
5

B
6

C
7

D
8

Solution

For R add ( 1,1) ( 2, 2) (3, 3) - 3
For S add (2, 1) (3, 2)- 2
For T add (1, 3) ( 3, 1) - 2
Thus the minimum number of ordered pairs to be added is 7.
Question 3
1 / -0

If $$A=\left\{2,4\right\}$$ and $$B=\left\{3,4,5\right\},$$ then
$$\left( A\cap B \right) \times \left( A\cup B \right)$$ is

A
$$\left\{(2,2),(3,4),(4,2),(5,4)\right\}$$

B
$$\left\{(2,3),(4,3),(4,5)\right\}$$

C
$$\left\{(2,4),(3,4),(4,4),(4,5)\right\}$$

D
$$\left\{(4,2),(4,3),(4,4),(4,5)\right\}$$

Solution

From given we get,
$$A\cap B=\{ 4\} ,A\cup B=\{ 2,3,4,5\}$$

$$\therefore \left( A\cap B \right) \times\left( A\cup B \right)$$$$=\left\{ (4,2),(4,3),(4,4),(4,5) \right\} $$

Ans. $$(d)$$.
Question 4
1 / -0

Let $$A=\left\{1,2,3,4,5\right\}, B=\left\{2,3,6,7\right\}.$$ Then the number of elements in $$\left( A\times B \right) \cap \left( B\times A \right)$$ is

A
$$18$$

B
$$6$$

C
$$4$$

D
$$0$$

Solution

Ans. $$(c)$$.
Common elements will be
$$\left\{ { (2,2),(3,3),(2,3),(3,2) } \right\} $$
Question 5
1 / -0

Let R be a relation from a set A to a set B then

A
$$R = A \cup B$$

B
$$ R = A \cap B$$

C
$$ R \subseteq A\times B$$

D
$$ R\subseteq B\times A$$

Solution

Let R be a relation from a set A to a set B then

$$R\subseteq A\times B$$
Question 6
1 / -0

The union of two equivalence relations is:

A
always an equivalence relation

B
reflexive and symmetric but needn't be transitive

C
reflexive and transitive but need not be symmetric

D
None of these

Solution

The union of two equivalence relation is not necessarily an equivalence relation.
Union of reflexive relation is reflexive,
Also, the union of symmetric relation is symmetric.
But the union of a transitive relation is not necessarily transitive.
Hence, the union of two equivalence relation is not equivalence.
Question 7
1 / -0

If $$n(A) = 2, n(B) = m$$ and the number of relation from $$A$$ to $$B$$ is $$64$$, then the value of $$m$$ is

A
$$6$$

B
$$3$$

C
$$16$$

D
$$8$$

Solution

The number of relations between sets can be calculated using $$ 2^{mn}$$ where m and n represent
the number of members in each set.
Given that $$ n(A) = 2 $$ and $$ n(B) = m $$
So,
$$ 2^{2m} = 64 $$
$$ 2^{2m} = 2^{6} $$
On comparing powers of 2 we get,
$$ 2m = 6 $$
$$ m = 3 $$
Question 8
1 / -0

If $$A$$ and $$B$$ are two sets containing four and two elements, respectively. Then the number of subsets of the set $$A\times B$$ each having at least three elements is

A
$$219$$

B
$$256$$

C
$$275$$

D
$$510$$

Solution

$$A=\left\{a,b,c,d\right\}$$
$$B=\left\{x,y\right\}$$
$$A\times B=2\times 4=8$$ Elements.
Total number of subsets of $$A\times B$$ having $$3$$ or more elements $$=2^8=256$$
$$\Rightarrow 256-(1$$ null set $$+$$ $$8$$ singleton set $$+$$ $$^8e_2$$having $$2$$ elements$$)$$
$$=256-1-8-28$$
$$=219$$
Question 9
1 / -0

Let $$R$$ be a relation such that $$R = \{(1,4), (3, 7), (4, 5), (4, 6), (7, 6) \}$$ then $$(R^{-1} oR)^{-1} =$$

A
$$\{(1, 1), (3, 3), (4, 4), (7, 7), (4, 7), (7, 4), (4, 3)\}$$

B
$$\{(1, 1), (3, 3), (4, 4), (7, 7), (4, 7), (7, 4) \}$$

C
$$\{(1, 1), (3, 3), (4, 4) \}$$

D
$$\phi$$

Solution
Question 10
1 / -0

If $$A$$ and $$B$$ are two sets, then $$A \times B = B \times A$$ if

A
$$A \subset B$$

B
$$B \subset A$$

C
$$A = B$$

D
None of these

Solution

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Re-Attempt Weekly Quiz Competition

Relations Test 6

Result

Relations Test 6

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Question 1

Reflexive

Symmetric

Transitive

None of these

Solution

Question 2

5

6

7

8

Solution

Question 3

$$\left\{(2,2),(3,4),(4,2),(5,4)\right\}$$

$$\left\{(2,3),(4,3),(4,5)\right\}$$

$$\left\{(2,4),(3,4),(4,4),(4,5)\right\}$$

$$\left\{(4,2),(4,3),(4,4),(4,5)\right\}$$

Solution

Question 4

$$18$$

$$6$$

$$4$$

$$0$$

Solution

Question 5

$$R = A \cup B$$

$$ R = A \cap B$$

$$ R \subseteq A\times B$$

$$ R\subseteq B\times A$$

Solution

Question 6

always an equivalence relation

reflexive and symmetric but needn't be transitive

reflexive and transitive but need not be symmetric

None of these

Solution

Question 7

$$6$$

$$3$$

$$16$$

$$8$$

Solution

Question 8

$$219$$

$$256$$

$$275$$

$$510$$

Solution

Question 9

$$\{(1, 1), (3, 3), (4, 4), (7, 7), (4, 7), (7, 4), (4, 3)\}$$

$$\{(1, 1), (3, 3), (4, 4), (7, 7), (4, 7), (7, 4) \}$$

$$\{(1, 1), (3, 3), (4, 4) \}$$

$$\phi$$

Solution

Question 10

$$A \subset B$$

$$B \subset A$$

$$A = B$$

None of these

Solution

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