Question

Give an example of a relation which is reflexive and symmetric but not transitive.

Answer 1

Hint: In this particular question assume any set (say A = $\left\{ {4,6,8} \right\}$), and assume any relation corresponding to the elements of assumed set A (say, R = $\left\{ {\left( {4,4} \right),\left( {6,6} \right),\left( {8,8} \right),\left( {4,6} \right),\left( {6,4} \right),\left( {6,8} \right),\left( {8,6} \right)} \right\}$), so use this concept to reach the solution of the question.

Complete step-by-step answer:
Let us consider a set A = $\left\{ {4,6,8} \right\}$
And a relation R = $\left\{ {\left( {4,4} \right),\left( {6,6} \right),\left( {8,8} \right),\left( {4,6} \right),\left( {6,4} \right),\left( {6,8} \right),\left( {8,6} \right)} \right\}$
Reflexive relation
If all the ordered pair elements of set A are in R then R is known to be called a reflexive relation.
Therefore, (a, a) $ \in $R, for all a$ \in $A.
So in set (A) all ordered pairs are (4, 4), (6, 6) and (8, 8) so all these ordered pairs are in set R so R is a reflexive relation.
Symmetric relation
If (a, b $ \in $A) such that (a, b) $ \in $ R then (b, a) $ \in $ R so this is called a symmetric relation.
For all (a, b$ \in $A)
So, R is also a symmetric relation.
So as we see that, $\left\{ {\left( {4,6} \right),\left( {6,4} \right),\left( {6,8} \right),\left( {8,6} \right)} \right\} \in R$, for all (4, 6, 8 $ \in $A)
So we can say that the relation R is a symmetric relation.
Transitive relation
If a, b, c $ \in $A such that (a, b) $ \in $ R and (b, c) $ \in $ R then (a, c) $ \in $ R so this is called a transitive relation.
So as we see that, $\left( {4,6} \right),\left( {6,8} \right) \in R$ for all a, b, c$ \in $A.
But $\left( {4.8} \right) \notin R$
So, R is not a transitive relation.
Hence R is a reflexive and symmetric but not transitive.
So this is the required answer.

Note: Whenever we face such types of problems the key point we have to remember is to have an adequate knowledge of sets and its properties. The definitions of all the three properties: reflexive, symmetric and transitive are written above, then one by one substituted with the assumed relation according to the definitions as above we will get the required answer.