Enquiry on Transitivity of a relation

Question

In transitive relation we know that if an element $a $ is related to $b $ and if $b$ is related to $c $ then $a $ is related to $c $. What I can't find any where is can $a $ be equal to $c $.??

Answer 1

Yes, if a relation $R$ is transitive and $a R b$ as well as $b R a$, then it must be $a R a$.

Formally, $R$ is transitive if it satisfies,

$$ \forall x\,.\, \forall y \,.\, \forall z \,.\, (xRy \wedge y R z) \rightarrow xRz \enspace. $$

No exclusions are made in the formula. Nowhere do we say, "if $x \neq y$" or anything of that sort. Hence you can substitute any element of the domain for any of the variables, even multiple times. Either the implication holds, or else $R$ is declared non-transitive.

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As an example, suppose we know that $R$ is transitive and symmetric, that is,

$$ \forall x \,.\, \forall y \,.\, x R y \rightarrow y R x \enspace. $$

Suppose we are also told that $R$ has this property:

$$ \forall x \,.\, \exists y \,.\, x R y \enspace. $$

That is, every element of the domain is related to at least one element of the domain. Then we can conclude that the relation is reflexive, which means that it is an equivalence relation. Suppose in fact that $a R b$. If $a = b$ then obviously $a R a$. Otherwise by symmetry $b R a$ and then by transitivity $a R a$.

Answer 2

Yes take example of $(1,2) \in R$ and $(2,1) \in R$ Then for transitivity $(1,1) \in R$.

Here a and c are both 1.