If Z is the set of integers. Then, the relation R = {(a,b):1+ab > 0} on Z is

see full answer

Your Exam Success, Personally Taken Care Of

1:1 expert mentors customize learning to your strength and weaknesses – so you score higher in school , IIT JEE and NEET entrance exams.

An Intiative by Sri Chaitanya

Reflexive and transitive but not symmetric

Symmetric and transitive but not reflexive

Reflexive and symmetric but not transitive

An equivalence relation

answer is C.

(Unlock A.I Detailed Solution for FREE)

Best Courses for You

JEE

NEET

Foundation JEE

Foundation NEET

CBSE

Detailed Solution

We are given that the relation is defined as R = {(a,b):1+ab > 0}
Now, let us check for the reflexive relation.
(1) A relation R is said to be reflexive if ARA satisfies the given relation.
Let us take the element in the relation R as (a,a)
By substituting the element in the given condition we get
⇒ 1+(a×a) > 0
⇒

a^{2}

> −1
Here we can see that the above equation satisfies for all a∈Z
So, we can conclude that the given relation is reflexive.
Now, let us check for symmetric relations.
(2) If ARB satisfies the relation and BRA satisfies the relation then that relation is said to be symmetric
Let us assume that the element (a,b) belongs to given relation R then by using the condition of relation we get
⇒ 1+ab > 0
⇒ 1+ba > 0
Here we can see that the element (b,a) also satisfies for all a,b∈Z
Therefore, we can conclude that the given relation is symmetric.
Now, let us check for transitive relation
(3) If ARB and BRC satisfies the relation then AR also satisfies the relation then that relation is said to be transitive.
Let us assume that the elements (a,b),(b,c) belongs to given relation then by using the condition we get
⇒1+ab>0
⇒ 1+bc > 0
Now, by adding the above two equation we get
⇒ 2+b(a+c) > 0
Here we can see that there is no further calculation to prove that 1+ac > 0
This means that the element (a,c) may or may not belong to a given relation R.
But we know that in order to call it a relation it should satisfy for all values
Therefore we can conclude that a given relation is not transitive.
Hence the given relation is reflexive and symmetric but not transitive.
So, option 3 is the correct answer.

Watch 3-min video & get full concept clarity