# Let S be the set of all sets and let R = {(A, B) : A ⊂ B)}, i.e., A is a proper subset of B. Show that R is (i) transitive (ii) not reflexive (iii) not symmetric.

Let R = {(A, B) : A B)}, i.e., A is a proper subset of B, be a relation defined on S.

Now,

Any set is a subset of itself, but not a proper subset.

(A,A) R A S

R is not reflexive.

Let (A,B) R A, B S

A is a proper subset of B

all elements of A are in B, but B contains at least one element that is not in A.

B cannot be a proper subset of A

(B,A) R

For e.g. , if B = {1,2,5} then A = {1,5} is a proper subset of B . we observe that B is not a proper subset of A.

R is not symmetric

Let (A,B) R and (B,C) R A, B,C S

A is a proper subset of B and B is a proper subset of C

A is a proper subset of C

(A,C) R

For e.g. , if B = {1,2,5} then A = {1,5} is a proper subset of B .

And if C = {1,2,5,7} then B = {1,2,5} is a proper subset of C.

We observe that A = {1,5} is a proper subset of C also.

R is transitive.

Thus, R is transitive but not reflexive and not symmetric.

Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Related Videos
Functional Equations - JEE with ease48 mins
Drift velocity & its relation with currentFREE Class
Relation between electric field and potential34 mins
Relation between Electric field & Potential45 mins
Interactive Quiz on Drift velocity & its relation with current38 mins
Interactive quiz on Relation between electric field and potential43 mins
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
view all courses