Relations Test 36

Result

Relations Test 36

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Question 1
1 / -0

If two sets $$A$$ and $$B$$ have $$ 99$$ elements in common, then the number of elements common to each of the sets $$A \times B$$ and $$B \times A$$ are

A
$$2^{99}$$

B
$$99^{2}$$

C
$$100$$

D
$$18$$

Solution

To find $$A \times B$$ we take one element from set $$A$$ and one from set $$B$$.
Given that $$99$$ elements are common to both set $$A$$ and set $$B$$.
Suppose these common elements are $$N_1, N_2, N_3,...N_{99}$$.
Select an ordered pair for $$A \times B$$ such that both are selected out of these common elements. Examples: $$\left(N_1,N_2 \right),\; \left(N_3,N_5 \right)$$
All these will also be elements of $$B \times A$$. Hence number of elements common to $$A \times B$$ and $$B \times A$$ is $$99 \times 99 = 99^2$$ ( first element in ordered pair can be selected in 99 ways; second element can also be selected in 99 ways)
$$n\left [ \left ( A\times B \right )\cap \left ( B\times A \right ) \right ]=n\left [ \left ( A\cap B \right )\cap \left ( B\cap A \right ) \right ]=(99)(99)=99^{2}$$
Question 2
1 / -0

Let A = {1, 2, 3}. Then number of relations containing (1, 2) and (1, 3) which are reflexive and symmetric but not transitive is

A
1

B
2

C
3

D
4

Solution
Question 3
1 / -0

If $$A=\left \{ 1,2,3 \right \} $$ and $$B=\left \{ 4,5,6 \right \}$$ then which of the following sets are relation from $$A$$ to $$B$$
(i) $$\displaystyle R_{1}=\left \{ (4,2) (2,6)(5,1)(2,4)\right \}$$
(ii) $$\displaystyle R_{2}=\left \{ (1,4) (1,5)(3,6)(2,6) (3,4)\right \}$$
(iii) $$\displaystyle R_{3}=\left \{ (1,5) (2,4)(3,6)\right \}$$
(iv) $$\displaystyle R_{4}=\left \{ (1,4) (1,5)(1,6)\right \}$$

A
$$\displaystyle R_{1},R_{2},R_{3}$$

B
$$\displaystyle R_{1},R_{3},R_{4}$$

C
$$\displaystyle R_{2},R_{3},R_{4}$$

D
$$\displaystyle R_{1},R_{2},R_{3},R_{4}$$

Solution

A relation from $$A$$ to $$B$$ will include elements from $$A\times B$$
Elements of $$A\times B$$ are
$$\{(1,4), (1,5) ,(1,6), (2,4), (2,5), (2,6), (3,4), (3,5), (3,6)\}$$
Now the relation $$R_{1}$$ consists of elements form both $$A\times B$$ and $$B\times A$$.
Thus $$R_{1}$$ is not a relation from $$A$$ to $$B.$$
Question 4
1 / -0

If n | m means that n is a factor of m, the relation | is

A
reflexive and symmetric

B
transitive and symmetric

C
reflexive, transitive and symmetric

D
reflexive, transitive and not symmetric

Solution
Question 5
1 / -0

The relation R {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)} on set A {1, 2, 3} is

A
reflexive but not symmetric

B
reflexive but not transitive

C
symmetric and transitive

D
neither symmetric nor transitive
Question 6
1 / -0

The relation "is subset of" on the power set P(A) of a set A is

A
symmetric

B
anti-symmetric

C
equivalence relation

D
reflexive
Question 7
1 / -0

The minimum number of elements that must be added to the relation R {(1, 2), (2, )} on the set of natural numbers so that it is an equivalence is

A
4

B
7

C
6

D
5

Solution
Question 8
1 / -0

Let $$R_1$$ and $$R_2$$ be equivalence relations on a set A, then $$R_1 \cup R_2$$ may or may not be

A
reflexive

B
symmetric

C
transitive

D
anti-symmetric

Solution
Question 9
1 / -0

Let N denote the set of all natural numbers and R a relation on N $$\times$$ N. Which of the following is an equivalence relation?

A
(a, b) R (c, d) if ad (b + c) = bc (a + d)

B
(a, b) R (c, d) if a + d = b + c

C
(a, b) R (c, d) if ad = bc

D
all the given

Solution
Question 10
1 / -0

Let A and B be finite sets containing m and n elements respectively. The number of relations that can be defined from A to B is

A
$$mn$$

B
$$2^{mn}$$

C
$$2^{m+n}$$

D
$$n^m$$

Solution