Question

Let \[A=\{1,\text{ }2,\text{ }3)\] and \[{{R}_{2}}=\{(2,2),(3,1),(1,3)\}\],
Find whether or not each of the relations \[{{R}_{2}}\] on A is
(i) reflexive (ii) symmetric (iii) transitive.

Answer 1

Hint: We will use the definitions of reflexive, symmetric and transitive relations to solve this question. A relation is a reflexive relation If every element of set A maps to itself. A relation in a set A is a symmetric relation if \[({{a}_{1}},{{a}_{2}})\in R\] implies that \[({{a}_{2}},{{a}_{1}})\in R\], for all \[{{a}_{1}},{{a}_{2}}\in A\]. A relation in a set A is a transitive relation if \[({{a}_{1}},{{a}_{2}})\in R\] and \[({{a}_{2}},{{a}_{1}})\in R\] implies that \[({{a}_{1}},{{a}_{3}})\in R\] for all \[{{a}_{1}},{{a}_{2}},{{a}_{3}}\in A\].

Complete step-by-step answer:
 Before proceeding with the question we should know about the concept of relations and different types of relations that are reflexive, symmetric and transitive relations.
A relation in set A is a subset of \[A\times A\]. Thus, \[A\times A\] is two extreme relations.
A relation in a set A is a reflexive relation if \[(a,a)\in R\], for every \[a\in A\].
A relation in a set A is a symmetric relation if \[({{a}_{1}},{{a}_{2}})\in R\] implies that \[({{a}_{2}},{{a}_{1}})\in R\], for all \[{{a}_{1}},{{a}_{2}}\in A\].
A relation in a set A is a transitive relation if \[({{a}_{1}},{{a}_{2}})\in R\] and \[({{a}_{2}},{{a}_{1}})\in R\] implies that \[({{a}_{1}},{{a}_{3}})\in R\] for all \[{{a}_{1}},{{a}_{2}},{{a}_{3}}\in A\].
A relation in a set A is an equivalence relation if R is reflexive, symmetric and transitive.
It is mentioned in the question that the relation \[{{R}_{2}}=\{(2,2),(3,1),(1,3)\}\] defined on the set A is {1, 2, 3). So according to the above definition the relation is not reflexive as (1, 1) and (3, 3) is absent.
Also \[(1,3)\in R\] and \[(3,1)\in R\], so the relation is symmetric.
Now \[(3,1)\in R\] and also \[(1,3)\in R\] but \[(3,3)\notin R\]. Hence the relation is not transitive.
Hence \[{{R}_{2}}=\{(2,2),(3,1),(1,3)\}\] is not reflexive, not transitive but symmetric.
Note: Remembering the definition of relations and the types of relations is the key here. We in a hurry can make a mistake in thinking it as a reflexive set but we have to check the definition by taking subsets of the given set A.