# Is it true that every relation which is symmetric and transitive is also reflexive ? Give reasons.

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Hint: - If you have to check whether a relation is reflexive or not in this question , first you have to assume a relation which is symmetric and transitive both then you have to check for reflexive(that means you have to check the number is in relation with itself or not).

“Complete step-by-step answer:”

Here we have to check that every relation which is symmetric and transitive is also reflexive or not.

So first you have to understand what is symmetric, reflexive, and transitive relation.

Symmetric relation: - If a is in relation with b then for symmetric relation b should be in relation with a.

Transitive relation: -If a is in relation with b and b is in relation with c then for transitive relation a should be in relation with c.

Reflexive relation: -If a number is in relation with itself then it is called a reflexive relation.

Consider the set I of the integers and a relation be defined as (aRb) if both a and b are odd.

Clearly aRb $ \Rightarrow $bRa that is if a and b are both odd then b and a are also both odd, so it is a symmetric relation.

Similarly, aRb and bRc implies aRc and hence transitive.

But this relation is not reflexive because $2 \in {\text{I}}$ but it can’t be in relation with 2 that is with itself to be in reflexive relation, because 2 is even number, and condition for to be in relation is number has to be a odd number.

Hence it is not true that every relation which is symmetric and transitive is also reflexive.

Note: -Whenever you get these types of questions the key concept of solving is you have first knowledge of all the relations like reflexive, symmetric and transitive and then you have to take an example which is symmetric and transitive both but not reflexive .

“Complete step-by-step answer:”

Here we have to check that every relation which is symmetric and transitive is also reflexive or not.

So first you have to understand what is symmetric, reflexive, and transitive relation.

Symmetric relation: - If a is in relation with b then for symmetric relation b should be in relation with a.

Transitive relation: -If a is in relation with b and b is in relation with c then for transitive relation a should be in relation with c.

Reflexive relation: -If a number is in relation with itself then it is called a reflexive relation.

Consider the set I of the integers and a relation be defined as (aRb) if both a and b are odd.

Clearly aRb $ \Rightarrow $bRa that is if a and b are both odd then b and a are also both odd, so it is a symmetric relation.

Similarly, aRb and bRc implies aRc and hence transitive.

But this relation is not reflexive because $2 \in {\text{I}}$ but it can’t be in relation with 2 that is with itself to be in reflexive relation, because 2 is even number, and condition for to be in relation is number has to be a odd number.

Hence it is not true that every relation which is symmetric and transitive is also reflexive.

Note: -Whenever you get these types of questions the key concept of solving is you have first knowledge of all the relations like reflexive, symmetric and transitive and then you have to take an example which is symmetric and transitive both but not reflexive .

Last updated date: 24th May 2023

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