Question

Let \[A = \left\{ {1,2,3} \right\}\] and \[R = \left\{ {\left( {1,2} \right),\left( {2,3} \right),\left( {1,3} \right)} \right\}\] be a relation on A. Then, R is
(a) neither reflexive nor transitive
(b) neither symmetric nor transitive
(c) transitive
(d) none of these

Answer 1

Hint:
Here we will use the concept of reflexive, transitive and symmetric. We will use the definition of each condition and check whether the condition satisfies the given relation or not. The condition, which will satisfy the relation, is the required answer.

Complete Step by step Solution:
Given relation is \[A = \left\{ {1,2,3} \right\}\] and \[R = \left\{ {\left( {1,2} \right),\left( {2,3} \right),\left( {1,3} \right)} \right\}\]
We will check each of the conditions of reflexive, transitive and symmetric for the given relation and conclude the type of the given relation.
First, we will check whether the relation is reflexive or not. We know that for a relation to be reflexive condition is that there must be \[\left( {a,a} \right) \in R\] and in our case \[\left( {1,1} \right),\left( {2,2} \right),\left( {3,3} \right)\] are missing from the relation. Therefore, the given relation is not reflexive.
Now we will check for the symmetry of the relation. We know that for a relation to be symmetric condition is that there must be \[\left( {a,b} \right) \in R\& \left( {b,a} \right) \in R\] and in our case \[\left( {2,1} \right),\left( {3,2} \right),\left( {3,1} \right)\] are missing from the relation. Therefore, the given relation is not symmetric as well.
Now we will check whether the relation is transitive or not. We know that for a relation to be transitive condition is that if \[\left( {a,b} \right) \in R\& \left( {b,c} \right) \in R\] there must be \[\left( {a,c} \right) \in R\]. In our case \[\left( {1,2} \right) \in R\& \left( {2,3} \right) \in R\] are in the relation and as well as its transitive function i.e. \[\left( {1,3} \right) \in R\] is there in the relation. Therefore, the given relation is transitive.
Hence, the given relation is transitive relation.

So, option C is the correct option.

Note:
Here, we need to know the basic condition of all the three properties i.e. reflexive, transitive and symmetric of a relation because it will make us easy to find whether the function belongs to it or not. A relation is a function which is the relation between the elements of the given array of elements. One-one functions are the functions that have their unique image in the domain whereas onto functions are the functions which have multiple images in the domain. If the function is not one-one then the function is generally known as many one function. One-one functions are generally known as injective functions and onto functions are generally known as the surjective functions. Bijective Functions are the functions which are both one-one and onto.