Question

Sherlin and Danju are playing Ludo at home during Covid-19. While rolling the dice, Sherlin’s sister Raji observed and noted the possible outcomes of the throw every time belongs to set {1, 2, 3, 4, 5, 6}. Let A be the set of players while B be the set of all possible outcomes.

A = {S, D}, B = {1, 2, 3, 4, 5, 6}

1. Let R ∶ B → B be defined by R = {(x, y ): y is divisible by x } is

a. Reflexive and transitive but not symmetric

b. Reflexive and symmetric and not transitive

c. Not reflexive but symmetric and transitive

d. Equivalence

2. Raji wants to know the number of functions from A to B. How many number of functions are possible?

a. 6²

b. 2⁶

c. 6!

d. 2¹²

3. Let R be a relation on B defined by R = {(1,2), (2,2), (1,3), (3,4), (3,1), (4,3), (5,5)}. Then R is

a. Symmetric

b. Reflexive

c. Transitive

d. None of these three

4. Raji wants to know the number of relations possible from A to B. How many numbers of relations are possible?

a. 6²

b. 2⁶

c. 6!

d. 2¹²

5. Let R : B → B be defined by R = {(1,1),(1,2), (2,2), (3,3), (4,4), (5,5),(6,6)}, then R is

a. Symmetric

b. Reflexive and Transitive

c. Transitive and symmetric

d. Equivalence

Answer 1

1. (a) Reflexive and transitive but not symmetric

Thus, R is reflexive and transitive but not symmetric.

2. (a) 6²

Given

A = {S, D}, B = {1, 2, 3, 4, 5, 6}

So, A has 2 elements, B has 6 elements

Numbers of functions from A to B = 6²

3. (d) None of these three

Given R = {(1, 2), (2, 2), (1, 3), (3, 4), (3, 1), (4, 3), (5, 5)}.

Check Reflexive

Here, (1, 1) ∉ R

∴ R is not reflexive

Check symmetric

To check whether symmetric or not,

If (x, y) ∉ R, then (y, x) ∉ R

Here (1, 2) ∉ R, but (2, 1) ∉ R

∴ R is not symmetric

Check transitive

To check whether transitive or not,

Thus, R is not reflexive, not symmetric and not transitive.

4. (d) 2¹²

Given

A = {S, D}, B = {1, 2, 3, 4, 5, 6}

Numbers of Relation from A to B

= 2^{Numbers of elements of A x Number of elements of B}

= 2^{2 × 6}

= 2¹²

5. (b) Reflexive and Transitive

Given R = {(1, 1), (1, 2), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}

Since R = {(1, 1), (1, 2), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}

Thus, R is reflexive and transitive.

Answer 2

1. (a) Reflexive and transitive but not symmetric

2. (a) 6²

3. (d) None of these three

4. (d) 212

5. (b) Reflexive and Transitive