Question

Test whether the relation \[{{R}_{3}}\] is
(i) Reflexive (ii) Symmetric and (iii) Transitive
where \[{{R}_{3}}\] on R defined by \[(a,b)\in {{R}_{3}}\Leftrightarrow {{a}^{2}}-4ab+3{{b}^{2}}=0\]

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.
Now let’s check for reflexivity. Assuming a to be an arbitrary element of \[{{R}_{3}}\].
Then, \[a\in {{R}_{3}}\Leftrightarrow {{a}^{2}}-4a\times a+3{{a}^{2}}=0\]. Hence it is reflexive.
Now moving on to symmetry. Let \[(a,b)\in {{R}_{3}}\Leftrightarrow {{a}^{2}}-4ab+3{{b}^{2}}=0\]. But \[{{b}^{2}}-4ba+3{{a}^{2}}\ne 0\] for all \[(a,b)\in R\]. Thus it is not symmetric.
Finally, checking for transitivity. Let \[(a,b)\in {{R}_{3}}\] and \[(b,c)\in {{R}_{3}}\]. So substituting \[(a,b)\in {{R}_{3}}\] we get \[{{a}^{2}}-4ab+3{{b}^{2}}\] and by substituting \[(b,c)\in {{R}_{3}}\] we get \[{{b}^{2}}-4bc+3{{c}^{2}}\]. And substituting \[(a,c)\in {{R}_{3}}\] we get \[{{a}^{2}}-4ac+3{{c}^{2}}\]. Hence it is not transitive.
Thus \[(a,b)\in {{R}_{3}}\Leftrightarrow {{a}^{2}}-4ab+3{{b}^{2}}=0\] is only reflexive.

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 symmetric set but on substituting (b, a) we get a different relation.