Question

Reflexive, transitive but not symmetric.

Answer 1

Hint: Here, we are required to show an example of a relation such that it is reflexive, transitive but not symmetric. We will form a set $A$ and make a relation ${R_1}$ on it such that it becomes an example of the given situation. We will use the definition of reflexive, symmetric and transitive to the relation.

Complete step-by-step answer:
We will first define the following three terms:
1.Reflexive Relation: A relation $R$ on a given set $A$ is reflexive if and only if \[\left( {a,a} \right) \in R\] where $a$ is an element in set $A$ such that $a \in A$. This is also known as ‘Self-Relation’ in simple terms.
2.Symmetric Relation: A relation $R$ on a given set $A$ is symmetric if it is given that $\left( {a,b} \right) \in R$ and we find that $\left( {b,a} \right) \in R$ where, $a,b \in A$.
3.Transitive Relation: A relation $R$ on a given set $A$is transitive if we find that $\left( {a,b} \right) \in R$ and $\left( {b,c} \right) \in R$ then, $\left( {a,c} \right) \in R$ where, $a,b,c \in A$.
Now, in this question, we have to show a relation $R$ on a set $A$ such that it is Reflexive, transitive but not symmetric.
Let ${R_1}$ be any relation on a given set $A$ such that:
Set $A = \left\{ {1,2,3} \right\}$
${R_1} = \left\{ {\left( {1,1} \right),\left( {2,2} \right),\left( {3,3} \right),\left( {1,3} \right),\left( {3,2} \right),\left( {1,2} \right)} \right\}$
Now, if we observe carefully then the above relation ${R_1}$ is reflexive because for every element $a \in A$, \[\left( {a,a} \right) \in R\] i.e. $\left\{ {\left( {1,1} \right),\left( {2,2} \right),\left( {3,3} \right)} \right\}$
Also, the above relation ${R_1}$ is transitive for instance, \[\left( {1,3} \right) \in R\] and \[\left( {3,2} \right) \in R\] hence, \[\left( {1,2} \right) \in R\]
Therefore, the given relation is transitive.
Now, as we can see, \[\left( {3,2} \right) \in R\] but \[\left( {2,3} \right) \notin R\]
Hence, this relation is not symmetric.
Therefore, this is an example of a relation which is Reflexive, transitive but not symmetric.

Note: In order to answer such types of questions, we should know the difference between reflexive, symmetric, and transitive relations as it will help us to identify which relation is symmetric and which one is not. Also, if any relation is reflexive, symmetric as well as transitive, it is known as an equivalence relation.