Question

Three relations \[{{R}_{1}}\], \[{{R}_{2}}\] and \[{{R}_{3}}\] are defined on a set \[A=\{a,\,b,c\}\] as follows:
\[{{R}_{1}}=\{(a,\,a),\,(a,b),(a,c),(b,b),(b,c),(c,a)(c,b),(c,c)\}\]
\[{{R}_{2}}=\{(a,\,a)\}\]
\[{{R}_{3}}=\{(b,\,c)\}\]
\[{{R}_{4}}=\{(a,b),(b,c),(c,a)\}\]
Find whether or not each of the relations \[{{R}_{1}}\], \[{{R}_{2}}\], \[{{R}_{3}}\], \[{{R}_{4}}\] 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:
(i) Reflexive: A relation is a reflexive relation If every element of set A maps to itself.
We will see \[{{R}_{1}}=\{(a,\,a),\,(a,b),(a,c),(b,b),(b,c),(c,a)(c,b),(c,c)\}\] first and here we can clearly see that (a, a), (b, b) and (c, c) belongs to \[{{R}_{1}}\]. Hence \[{{R}_{1}}\] is reflexive.
Now we move on to \[{{R}_{2}}=\{(a,\,a)\}\] and here we can clearly see that (a, a) belongs to \[{{R}_{2}}\]. Hence \[{{R}_{2}}\] is Reflexive.
Now we move on to \[{{R}_{3}}=\{(b,\,c)\}\] and here we can clearly see that (b, b), (c, c) does not belongs to \[{{R}_{3}}\]. Hence \[{{R}_{3}}\] is not reflexive.
Now we move on to \[{{R}_{4}}=\{(a,b),(b,c),(c,a)\}\] and here we can clearly see that (a, a), (b, b), (c, c) does not belongs to \[{{R}_{4}}\]. Hence \[{{R}_{4}}\] is not reflexive.

(ii) Symmetric: 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 \[{{a}_{1}},{{a}_{2}}\in A\].
We will see \[{{R}_{1}}=\{(a,\,a),\,(a,b),(a,c),(b,b),(b,c),(c,a)(c,b),(c,c)\}\] first and here we can clearly see that (b, a) does not belongs to \[{{R}_{1}}\]. Hence \[{{R}_{1}}\] is not symmetric.
Now we move on to \[{{R}_{2}}=\{(a,\,a)\}\] and here we can clearly see that (a, a) belongs to \[{{R}_{2}}\]. Hence \[{{R}_{2}}\] is Symmetric.
Now we move on to \[{{R}_{3}}=\{(b,\,c)\}\] and here we can clearly see that (c, b) does not belongs to \[{{R}_{3}}\]. Hence \[{{R}_{3}}\] is not symmetric.
Now we move on to \[{{R}_{4}}=\{(a,b),(b,c),(c,a)\}\] and here we can clearly see that (b, a) does not belongs to \[{{R}_{4}}\]. Hence \[{{R}_{4}}\] is not symmetric.

(iii) Transitive: 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 \[{{a}_{1}},{{a}_{2}},{{a}_{3}}\in A\].
We will see \[{{R}_{1}}=\{(a,\,a),\,(a,b),(a,c),(b,b),(b,c),(c,a)(c,b),(c,c)\}\] first and here we can clearly see that (a, b), (b, c) and also (a, c) belongs to \[{{R}_{1}}\]. Hence \[{{R}_{1}}\] is not Transitive.
Now we move on to \[{{R}_{2}}=\{(a,\,a)\}\] and it is clearly a transitive relation since there is only one element in it.
Now we move on to \[{{R}_{3}}=\{(b,\,c)\}\] and here it has only two elements. Hence it is Transitive.
Now we move on to \[{{R}_{4}}=\{(a,b),(b,c),(c,a)\}\] and here we can clearly see that (a, c) does not belongs to \[{{R}_{4}}\]. Hence \[{{R}_{4}}\] is not Transitive.

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 \[{{R}_{3}}\] as a symmetric set but we have to check the definition by taking subsets of the given set A.