Q. 3 A4.2( 10 Votes )

Test whether the following relations R1, R2 and R3 are (i) reflexive (ii) symmetric and (iii) transitive :

R1 on Q0 defined by (a, b) ϵ R1 a = 1/b

Answer :

Here, R1, R2, R3, and R4 are the binary relations.

So, recall that for any binary relation R on set A. We have,

R is reflexive if for all x A, xRx.

R is symmetric if for all x, y A, if xRy, then yRx.

R is transitive if for all x, y, z A, if xRy and yRz, then xRz.

So, using these results let us start determining given relations.

We have

R1 on Q0 defined by (a, b) R1

Check for Reflexivity:

a, b Q0,

(a, a), (b, b) R1 needs to be proved for reflexivity.

If (a, b) R1

Then, …(1)

So, for (a, a) R1

Replace b by a in equation (1), we get

But, we know

(a, a) R1

So, a Q0, then (a, a) R1

R1 is not reflexive.

Check for Symmetry:

If (a, b) R1

Then, (b, a) R1

a, b Q0

If (a, b) R1

We have, …(2)

Now, for (b, a) R1

Replace a by b & b by a in equation (2), we get

(b, a) R2

So, if (a, b) R1, then (b, a) R1

a, b Q0

R1 is symmetric.

Check for Transitivity:

If (a, b) R1 and (b, c) R1

We need to eliminate b.

We have

Putting in , we get


(a, c) R1

So, if (a, b) R1 and (b, c) R1, then (a, c) R1

a, b, c Q0

R1 is not transitive.

Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Related Videos
Functions - 0152 mins
Quick Revision of Types of Relations59 mins
Range of Functions58 mins
Functions - 0648 mins
Functions - 1156 mins
Battle of Graphs | various functions & their Graphs48 mins
Some standard real functions61 mins
Functions - 0947 mins
Quick Recap lecture of important graphs & functions58 mins
Range of Quadratic/quadratic & linear/Linear functions45 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