Test Name: RELATION

Board: CBSE Standard: 12th Subject: Mathematics

AND FUNCTION

Student Name:__________________ Section:________ Roll No.:__________

Time: 2 Minutes Worksheet

Section B
3 Marks
Attempt any 5 questions out of 10.

1: Check whether the relation R in the set Z of integers defined as R = {(a, b) : a + b is divisible by 2} is
reflexive, symmetric or transitive. Write the equivalence class containing 0 i.e. [0].

2: Prove that the relation R on Z, defined by R = {(x, y) : (x – y) is divisible by 5} is an equivalence relation.

3: Show that the relation R on R defined as R = {(a, b) : a ≤ b}, is reflexive, and transitive but
not symmetric.

4: Check whether the relation R in the set R of real numbers, defined by R = {(a, b) : 1 + ab > 0}, is
reflexive, symmetric or transitive.

5: Show that the relation R in the set N × N defined by (a, b) R (c, d) if a2 + d2 = b2 + c2 ∀ a, b, c, d ∈ N, is

an equivalence relation.

Section C
5 Marks
Attempt any 5 questions out of 10.

1: 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}

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

2: 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}

Raji wants to know the number of relations possible from A to B. How many numbers of
relations are possible?
3: 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}

Let R be a relation on B defined by R = {(1, 2), (2, 2), (1, 3), (3, 4), (3, 1), (4, 3), (5, 5)}. Is R an
equivalence relation?

OR

Show that a relation, R : B → B be defined by R = {(x, y) : y is divisible by x} is a reflexive and

transitive but not symmetric.

4: An organization conducted bike race under two different categories-Boys and Girls. There
were 28 participants in all. 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 } and G = {g1 , g2 }, where B represents the set of Boys selected and G be
the set of Girls selected for the final race.
 Based on the above information, answer the following questions:

How many relations are possible from B to G ?

5: An organization conducted bike race under two different categories-Boys and Girls. There
were 28 participants in all. 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 } and G = {g1 , g2 }, where B represents the set of Boys selected and G be
the set of Girls selected for the final race.

Based on the above information, answer the following questions:

Among all the possible relations from B to G, how many functions can be formed from B to
G?

6: An organization conducted bike race under two different categories-Boys and Girls. There
were 28 participants in all. 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 } and G = {g1 , g2 }, where B represents the set of Boys selected and G be
the set of Girls selected for the final race.
 Based on the above information, answer the following questions:

Let R : B → B be defined by R = {(x, y) : x and y are students of the same sex}. Check if R is
an equivalence relation.

OR

A function f : B → G be defined by f = {(b1 , g1 ), (b2 , g2 ), (b3 , g1 )}.

Check if f is bijective. Justify your answer.

7: Show that the function f : ℝ → {x ∈ ℝ : –1 < x < 1) defined by

x
f (x) =
1 + |x|

, x ∈ ℝ is one-one and onto function.

8: A function f : R – {–1, 1} → R is defined by:

x
f (x) =
2
x − 1

(i) Check if f is one-one.

(ii) Check if f is onto.

Show your work.

9: Consider f : R+ → [– 5, ∞) given by f(x) = 9x2 + 6x – 5 where R+ is the set of all non-negative

real numbers. Prove that f is one-one and onto function.
10: A function f : [– 4, 4] → [0, 4] is given by f(x) =
2
√16 − x

. Show that f is an onto function but not one-one function. Further, find all possible values of 'a' for
which f(a) =

√7