Case study Chapter 1 (Relation and Function)

Case study 4:- Read the following answer the question(Case study relation and function 4)

Student of gade 9,planned to plant saplings along straight lines, parallel to each other to one side of the play ground ensuring that they had enough play area. Let us assume that they planted one of the rows of the saplings along the line y = x – 4. Let L be the set of all lines which are parallel on the ground and R be a relation on L.

(i) Let relation R be defined by $R= \{(L_1, L_2): L_1||L_2 \text{ where } L_1,L_2 \in L\}$ Then R is ______ Relation

(a) Equivalence (b) Only Reflexive

(ii) Let $R =\{(L_, L_2): L_1\perp L_2 \text{ where } L_1, L_2 \in L\}$ which of the following is true ?

(a) R is Symmetric but neither reflexive nor transitive

(b) R is Reflexive and transitive but not Symmetric.

(d) R is equivalence relation.

(iii) The function f:R → R defined by f(x) = x – 4 is

(a) Bijective (b) Surjective but not injective

(d) Neither Surjective nor Injective

(iv) Let f:R → R be defined by f(x) = x – 4. then the range of f(x) is

(a) R (b) Z

(v) Let $R = \{(L_1, L_2): L_1 || L_2 \text{ and } L_1:y = x-4\}$ then which of the following can be taken as $L_2$

(a) 2x – 2y + 5 = 0 (b) 2x + y = 5

Solution ;(i) Answer (a)

Given relation R defined by

$R= \{(L_1, L_2): L_1||L_2 \text{ where } L_1,L_2 \in L\}$

Reflexive: Let $L_1 \in L \Rightarrow L_1||L_1$

$\Rightarrow (L_1, L_2) \in R$

It is reflexive

Symmetric: Let $L_1,L_2 \in L$ Such that

$(L_1,L_2) \in R \Rightarrow L_1||L_2$

$\Rightarrow L_2||L_1 \Righattrrow (L_2,L_1) \in R$

It is Symmetric

Transitive: Let $L_1,L_2,L_3 \in L$

$(L_1,L_2) \inR \text{ and } (L_2,L_3) \in R$

$\Rightarrow L_1||L_2 \text{ and } L_2||L_3$

$\Rightarrow L_1||L_3 \Rightarrow (L_1, L_3)\in R$

It is transitive.

Hence R is an equivalence relation.

(ii) Answer (a)

Given relation R defined by $R =\{(L_, L_2): L_1\perp L_2 \text{ where } L_1, L_2 \in L\}$

Reflexive: Since every line is not perpendicular to itself.

$\therefore (L_1,L_1) \notin R$

It is nor reflexive.

Symmetric: Let $L_1,L_2 \in L$

$\Rightarrow L_1 \perp L_2 \Rightarrow L_2 \perp L_1$

$\Rightarrow (L_2, L_1) \in R$

It is symmetric.

Transitive: Let $L_1, L_2,L_3 \in L$

Let $(L_1, L_2) \in R \text{ and } (L_2,L_3) \in R$

$\Rightarrow L_1\perp L_2 \text{ and } L_2 \per L_3$

$\Rightarrow L_1\perp L_3$

$\Rightarrow (L_1,L_3) \notin R$

∴ It is not Transitive

Hence relation R is symmetric but neither reflexive nor transitive.

(iii) Answer (a)

Given function f:R → R defined by f(x) = x – 4

Injective: Let $x_1,x_2 \in R$ such that $x_1 \neq x_2$

$\Rightarrow x_1 \neq x_2 \Rightarrow f(x_1) \neq f(x_2)$

∴ It is injective.

Surjective: Let y = x – 4 ⇒ x = y + 4

For every y ∈ R there exists x = y + 4 ∈ R.

i.e. Co- domain = Range

∴ It is surjective.

Hence given function is bijective.

(iv) Answer (a)

Given function f:R → R defined by f(x) = x – 4

Let y = f(x) ⇒ y = x – 4 ⇒ x = y + 4

⇒ x ∈ R ⇒ y ∈ R

∴ Range of f(x) is R (Set of real numbers).

(v) Answer (a)

2x – 2y + 5 = 0 is parallel to y = x – 4.

Some other case study problem

Case study 1:- Read the following and answer the question:(Case study relation and function 1)

A general election of Lock sabha is a gigantic exercise. About 311 million people were eligible to vote and voter turnout was about 67%, the highest ever

Let I be the set of all citizens of India who were eligible to exercise their voting right in general electio held in 2019. A relation ‘R’ is defined on I as follows:

$R = [(V_1, V_2): V_1,V_2 \in I$ and both use their voting right in general election – 2019]

Solution: For solution click here

Case study 2:- Sherlin and Danju are playing Ludo at home during COVID- 19. While rolling the dice, Sherlin’s sister Raji cbserved 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.(Case study relation and function 2)

Case study relation and function 2 — Sherlin and Danju are playing Ludo at home during COVID- 19

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

Based on the above information answer the question.

Solution: For solution click here

Case Study 3:- Read the following and answer the question:(Case study relation and function 3)

An organization conducted bike race under 2 different categories- boys and girls. In all, 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 from two sets B and G with these participants for his college project.

Let $B = {b_1, b_2, b_3}$ and $G = {g_1,g_2}$ where B represents the set of boys selected and G the set of girls who were selected for the final race.