Relations

A classroom teacher is keen to assess the learning of her students the concept of "relations" taught to them. She writes the following five relations each defined on the set

A%20%3D%20%5C%7B1%2C%202%2C%203%5C%7D

R_1%20%3D%20%5C%7B(2%2C%203)%2C%20(3%2C%202)%5C%7D%2C%20%5Cquad%20R_2%20%3D%20%5C%7B(1%2C%202)%2C%20(1%2C%203)%2C%20(3%2C%202)%5C%7D%2C%20%5Cquad%20R_3%20%3D%20%5C%7B(1%2C%202)%2C%20(2%2C%201)%2C%20(1%2C%201)%5C%7D%2C

R_4%20%3D%20%5C%7B(1%2C%201)%2C%20(1%2C%202)%2C%20(3%2C%203)%2C%20(2%2C%202)%5C%7D%2C%20%5Cquad%20R_5%20%3D%20%5C%7B(1%2C%201)%2C%20(1%2C%202)%2C%20(3%2C%203)%2C%20(2%2C%202)%2C%20(2%2C%201)%2C%20(2%2C%203)%2C%20(3%2C%202)%5C%7D.

The students are asked to answer the following questions about the above relations:
(i) Identify the relation which is reflexive, transitive but not symmetric.
\ (ii) Identify the relation which is reflexive and symmetric but not transitive.
(iii) Identify the relations which are symmetric but neither reflexive nor transitive.
OR
(iii) What pairs should be added to the relation

R_2

to make it an equivalence relation?

(i) Identify the relation which is reflexive, transitive but not symmetric:
- Reflexive: A relation

R

is reflexive if for all

a%20%5Cin%20A

(a%2C%20a)%20%5Cin%20R

.
- Transitive: A relation

R

is transitive if for all

(a%2C%20b)%20%5Cin%20R

and

(b%2C%20c)%20%5Cin%20R

(a%2C%20c)%20%5Cin%20R

.
- Symmetric: A relation

R

is symmetric if for all

(a%2C%20b)%20%5Cin%20R

(b%2C%20a)%20%5Cin%20R

.
Now, let's check each relation for reflexivity, transitivity, and symmetry.
-

R_1%20%3D%20%5C%7B(2%2C%203)%2C%20(3%2C%202)%5C%7D

:
- Not reflexive (missing

(1%2C%201)

(2%2C%202)

(3%2C%203)

).
- Not transitive because we don't have

(2%2C%202)

(3%2C%203)

for transitivity.
- Not symmetric since

(2%2C%203)

is in

R_1

, but

(3%2C%202)

is not.
-

R_2%20%3D%20%5C%7B(1%2C%202)%2C%20(1%2C%203)%2C%20(3%2C%202)%5C%7D

:
- Not reflexive (missing

(2%2C%202)

(3%2C%203)

).
- Transitive: It is transitive since if we have

(1%2C%202)

and

(2%2C%203)

, we also have

(1%2C%203)

, and similarly for other combinations.
- Not symmetric, because

(1%2C%202)

is in

R_2

, but

(2%2C%201)

is not.
-

R_3%20%3D%20%5C%7B(1%2C%202)%2C%20(2%2C%201)%2C%20(1%2C%201)%5C%7D

:
- Reflexive because it includes

(1%2C%201)

(2%2C%202)

, and

(3%2C%203)

.
- Symmetric because for every pair

(a%2C%20b)

(b%2C%20a)

is also present.
- Not transitive because there is no

(1%2C%203)

, which makes it not transitive.
Thus,

R_3

is reflexive and symmetric but not transitive.
(ii) Identify the relation which is reflexive and symmetric but not transitive:
From the analysis above,

R_3

is reflexive and symmetric but not transitive.
(iii) Identify the relations which are symmetric but neither reflexive nor transitive:
-

R_1

is symmetric but neither reflexive nor transitive.
-

R_5

is symmetric but neither reflexive nor transitive.
OR
(iii) What pairs should be added to the relation

R_2

to make it an equivalence relation? To make

R_2%20%3D%20%5C%7B(1%2C%202)%2C%20(1%2C%203)%2C%20(3%2C%202)%5C%7D

an equivalence relation, it needs to be reflexive, symmetric, and transitive. - Reflexive: We need to add

(1%2C%201)

(2%2C%202)

, and

(3%2C%203)

. - Symmetric: Add

(2%2C%201)

and

(3%2C%201)

(since

(1%2C%202)

(1%2C%203)

are already present, but their reverse pairs are missing). - Transitive: Ensure the necessary transitive pairs are present, which may be covered after adding the missing symmetric pairs. Thus, the pairs to be added are:

(1%2C%201)%2C%20(2%2C%202)%2C%20(3%2C%203)%2C%20(2%2C%201)%2C%20(3%2C%201).

High-Yield Trend

Chapter Questions
2 MCQs

Official Solution

Official Solution

Relations

High-Yield Trend

Chapter Questions 2 MCQs

Official Solution

Official Solution

Chapter Questions
2 MCQs