Question

An organization conducted bike race under 2 different categories-boys and girls. Totally there were 250 participants. Among all of them finally three from Category 1 and two from Category 2 were selected for the final race. Ravi forms two sets B and G with these participants for his college project.

Let B = {b₁, b₂, b₃} G={g₁, g₂} where B represents the set of boys selected and G the set of girls who were selected for the final race. 

Ravi decides to explore these sets for various types of relations and functions

1. Ravi wishes to form all the relations possible from B to G. How many such relations are possible?

a. 2⁶

b. 2⁵

c. 0

d. 2³

2. Let R: B→B be defined by R = {(x, y): x and y are students of same sex}, Then this relation R is_______

a. Equivalence

b. Reflexive only

c. Reflexive and symmetric but not transitive

d. Reflexive and transitive but not symmetric

3. Ravi wants to know among those relations, how many functions can be formed from B to G?

a. 2² 

b. 2¹²

c. 3²

d. 2³

4. Let R : B → G be defined by R = { (b₁, g₁), (b₂, g₂),(b₃, g₁)}, then R is__________

a. Injective

b. Surjective

c. Neither Surjective nor Injective

d. Surjective and Injective

5. Ravi wants to find the number of injective functions from B to G. How many numbers of injective functions are possible?

a. 0

b. 2!

c. 3!

d. 0!

Answer 1

1. (a) 2⁶

Given

B = {b₁, b₂, b₃} G = {g₁, g₂}

Numbers of Relation from B to G

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

= 2^{3 x 2}

= 2⁶

2. (a) Equivalence

R = {(x, y): x and y are students of same sex}.

Since R is reflexive, symmetric and transitive

∴ R is an Equivalence relation.

3. (d) 2³

Given

B = {b₁, b₂, b₃} G = {g₁, g₂}

So, B has 3 elements, G has 2 elements

Numbers of functions from B to G = 2 × 2 × 2

= 2³

4. (b) Surjective

Given B = {b₁, b₂, b₃}, G = {g₁, g₂}

And,

R = {(b₁, g₁), (b₂, g₂), (b₃, g₁)}

So, our relation looks like

5. (a) 0

In a function,

Every element of set B will have an image.

Every element of set B will only one image in set G

For injective functions

All elements of set G should have a unique pre-image

Which is not possible

∴ Number of Possible injective function = 0

Answer 2

1. (a) 2⁶

2. (a) Equivalence

3. (d) 2³

4. (b) Surjective

5. (a) 0