Question

Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6) as \[R=\{(a,b):b=a+1\}\] is reflexive, symmetric or 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 defined on the set A is {1, 2, 3, 4, 5, 6) as \[R=\{(a,b):b=a+1\}\]. So according to the above definition the relation is not reflexive as a is not equal to a+1 for any \[a\in A\].
Also \[3=2+1\] but \[2\ne 3+1\] that is \[(2,3)\in R\] but \[(3,2)\notin R\], so the relation is not symmetric.
Now \[(2,3)\in R\] and also \[(3,4)\in R\] but \[(2,4)\notin R\] since \[3=2+1\], \[4=3+1\] but \[4\ne 2+1\]. Hence the relation is not transitive also.
Hence \[R=\{(a,b):b=a+1\}\] is not reflexive, not symmetric and neither transitive.

Note: We in a hurry can make a mistake in thinking it as a transitive set and as a reflexive set but we have to check the definition by taking subsets of the given set A. A relation is a reflexive relation If every element of set A maps to itself. 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 \[{{a}_{1}},{{a}_{2}},{{a}_{3}}\in A\].