Question

Let \[R = \left\{ {\left( {3,3} \right)\left( {6,6} \right)\left( {9,9} \right)\left( {12,12} \right)\left( {6,12} \right)\left( {3,9} \right)\left( {3,12} \right)\left( {3,6} \right)} \right\}\;\] be a relation on the set \[A = \left\{ {3,6,9,12} \right\}.\] The relation is
A) reflexive and transitive only
B) reflexive only
C) an equivalence relation
D) reflexive and symmetric only

Answer 1

Hint: To solve this problem, we are given a relation on the set. We will check the relation of which type. At first, we will check whether the relation is reflexive is not, by using the condition of reflexive relation. Then we will check whether the relation is symmetric is not, by using the condition of symmetric relation. Similarly, we will check whether the relation is transitive or not. Then after getting what types of relation is formed, we will mark the correct multiple choice out of four options given to us.

Complete step-by-step answer:
We have been given that \[R = \left\{ {\left( {3,3} \right)\left( {6,6} \right)\left( {9,9} \right)\left( {12,12} \right)\left( {6,12} \right)\left( {3,9} \right)\left( {3,12} \right)\left( {3,6} \right)} \right\}\;\] is a relation on the set \[A = \left\{ {3,6,9,12} \right\}.\] We need to tell what type of relation it will form.
So, \[A = \left\{ {3,6,9,12} \right\},\] and R is a relation on set A.
We know that, if relation is reflexive, then \[(a,a) \in R,\] for every \[a \in A.\] Here, we can see that R is reflexive on set A, as every element of set A is related, i.e., (3,3), (6,6), (9,9), (12,12).
Now, we know that if relation is symmetric, then \[(a,b) \in R\] then, \[(b,a) \in R.\] Here, we can see that, R is not symmetric, as $(3,9) \in R$ but $(9,3) \notin R.$
Now, we know that if relation is transitive, then \[(a,b) \in R\] and $(b,c) \in R$ then, \[(a,c) \in R.\] Here, we can see that, R is transitive, as $(3,6) \in R,(6,12) \in R$ and also $(3,12) \in R.$
So, R is reflexive and transitive. Thus, option (A) reflexive and transitive only, is correct.
So, the correct answer is “Option A”.

Note: Equivalence relation is the relation which is reflexive, symmetric and also transitive. Here, the relation is only reflexive and transitive, that’s why we haven’t chosen the option equivalence relation. So, only option (A) is correct.