10th samacheer kalvi maths chapter 1 Relations and Functions, Exercise 1.2 questions with answer

Example 1.4 Let A = {3,4,7,8} and B = {1,7,10}. Which of the following sets are relations from A to B?

(i) R1 = {(3,7),(4,7),(7,10),(8,1)}

Solution

At first we need to find A x B and if all elements in R1 are from A x B then R1 is relation from A to B

                A x B =  {3,4,7,8} x  {1,7,10}

                          = {(3,1),(3,7),(3,10),(4,1),(4,7),(4,10),(7,1),(7,7),(7,10),(8,1),(8,7),(8,10)}

On comparing A x B and R1, all elements in R1 are belongs to A x B so, R1 is relation from A to B

            i.e R1⊆AxB. Thus, R1 is the relation from  A to B

(ii)R2 = {(3,1),(4,12)}

Solution 

Comparing AxB and R2, (4,12)∉AxB, so R2 is not a relation from A to B

(iii)R3 = {(3,7),(4,10),(7,7),(7,8),(8,11),(8,7),(8,10)}

Solution

Comparing AxB and R3, (7,8)∉AxB, hence R3 is not a relation from A to B

Example 1.5 The arrow diagram shows (Fig.1.10) a relationship between the sets P and Q. Write the relation in (i) Set builder form (ii) Roster form (iii) What is the domain and range of R.

Solution

(i)Set builder form of R = {(x,y) |y = x-2, x∈P, y∈Q}

(ii) Roster form R = {(5,3),(6,4),(7,5)}

(iii) Domain of R = {5,6,7}

                                                 Range of R = {3,4,5}

Exercise 1.2
1.Let A = {1,2,3,7} and B = {3,0,-1,7}, which of the following are relation from A to B?

(i)R1 = {(2,1),(7,1)}

Solution 

At first we need to find A x B and if all elements in R1 are from A x B then R1 is relation from A to B

            AxB = {1,2,3,7} x {3,0,-1,7}

                    = {(1,3),(1,0),(1,-1),(1,7),(2,3),(2,0),(2,-1),(2,7),(3,3),(3,0),(3,-1),(3,7),(7,3),                               (7,0),(7,-1),(7,7)}

                    Both (2,1) and (7,1) are not belongs to AxB

   Hence, R1 is not a relation from A to B

(ii)R2 = {(-1,1)}

Solution

Here, (-1,1) does not belongs to AxB

Hence, R2 is not a relation from A to B

(iii)R3 = {(2,-1),(7,7),(1,3)}

Solution

All elements in R3 belongs to AxB

Hence, R3 is the relation from A to B

(iv)R4 = {(7,-1),(0,3),(3,3),(0,7)}

Solution

Here (0,3) and (0,7) are not belongs to AxB

Hence, R4 is not a relation from A to  B

2.Let A = {1,2,3,4,5,6......,45} and R be the relation defined as "is square of a number" on A. Write R as  a subset of A x A. Also, find the domain and range of R. 

Solution

Given, A = {1,2,3,4,5,6......,45}

Relation is given as "Square of a number, 

So, R = {(1,1),(2,4),(3,9),(4,16),(5,25),(6,36)}

Hence, Domain of R = {1,2,3,4,5,6}

            Range of R = {1,4,9,16,25,36}

3.A relation R is given by the set {(x,y) / y = x+3, x∈{0,1,2,3,4,5}}. Determine its domain and Range. 

Solution

Since the relation is given, we have formula for y (y =x+3) and the values of x (0,1,2,3,4,5). Here we need to just put values of x in y = x+3.

Hence from above, R = {(0,3),(1,4),(2,5),(3,6),(4,7),(5,8)}

Domain of R = {0,1,2,3,4,5}

Range of R = {3,4,5,6,7,8}

4.Represent each of the given relations by (a) an arrow diagram, (b) a graph and (c) a set in roster form, wherever possible. 

(i){(x,y)| x = 2y, x ∈ {2,3,4,5}, y ∈ {1,2,3,4}}}

Solution 

Since the relation is given, we have formula for y (y =x/2) and the values of x (2,3,4,5). Here we need to just put values of x in y = x/2.

Given, x = 2y

By rearranging the above equation we can get the equation for y, y = x/2.

Since we get values for only when x = 2 and x = 4

a) Arrow diagram 

b) Graph

c) Roster form

    Roster form = {(2,1),(4,2)}

(ii) {(x,y)| y = x+3, x,y are natural numbers < 10}

Solution

Since the relation is given, we have formula for y (y =x+3) and the values of x (1,2,3,4,5,6,7,8,9) . Here we need to just put values of x in y = x+3.

Given, x = {1,2,3,4,5,6,7,8,9} (Natural numbers less than 10)

            y = {1,2,3,4,5,6,7,8,9} (Natural numbers less than 10)

a) Arrow Diagram

b)Graph

We have, R = {(1,4),(2,5),(3,6),(4,7),(5,8),(6,9)}

c) Roster form

     Roaster form = {(1,4),(2,5),(3,6),(4,7),(5,8),(6,9)}

5. A company has four categories of employees given by Assistants (A), Clerks (C), Managers (M) and an Executive Officer (E). The company provide ₹10,000, ₹25,000, ₹50,000 and ₹1,00,000 as salaries to the people who work in the categories A, C, M and E respectively. If A1, A2, A3, A4 and A5 were Assistants; C1, C2, C3, C4 were Clerks; M1, M2, M3 were managers and E1, E2 were Executive officers and if the relation R is defined by xRy, where x is the salary given to person y, express the relation R through an ordered pair and an arrow diagram

Solution

Given, Assistants (A) = {A1, A2, A3, A4,A5}

              Clerks (C) = {C1, C2, C3, C4}

            Managers (M) = {M1, M2, M3}

 Executive Officers (E) = {E1,E2}

           Salary for Assistants = 10000

          Salary for Clerks = 25000

            Salary for Managers = 50000

           Salary for Executive Officers = 100000

Roster form = {(10000,A1),(10000,A2),(10000,A3),(10000,A4),(10000,A5),(25000,C1),                                (25000,C2),  (25000,C3),(25000,C4),(50000,M1),(50000,M2),(50000,M3),                             (100000,E1),(100000,E2)}

Arrow diagram can be drawn as follows,