Must composition relation is transitive?

Question

I am confused by the transitive relation of the composition relation, this is the question.

$A$ and $B$ are transitive binary relation on set $X$, then $B\circ A$ is transitive.

Why this statement is false, since the definition of composition relation is:

$R\circ S$ is a compostional function, if $(a,c) \in R\circ S $ then there must exist a $b$ in set X such that $a R b$ and $bSc$.

In this way, the composition relation is transitive, but the correct answer is false, could someone give me an example, math or real-life example either is ok.

Answer 1

While it is true that $(a,c) \in R \circ S$ if and only if there is a $b$ such that $aRb$ and $bSc$, $R \circ S$ does not retain transitivity.

Here is an example : On the set $\{1,2,3,4,5\}$,we have the following two relations $R = \{(1,2),(3,4)\}$ and another $S = \{(2,3),(4,5)\}$. See that both $R$ and $S$ are transitive(vacuously i.e. there are no cases to check) but $R \circ S = \{(1,3),(3,5)\}$ which is not transitive.

Intuitively too, there is no reason why the composition of two relations must be transitive. If $(a,c),(c,e) \in R \circ S$, then there exist $b,d$ such that $(a,b),(c,d) \in R$, $(b,c),(d,e) \in S$. From merely this conclusion you can see that there is no reason why there must be some other $f$ such that $(a,f) \in R$ and $(f,e) \in S$.

Answer 2

Let's try to construct a counterexample.

Let's say $R = \{(a,b),(c,d)\}$ and $S=\{(b,c),(d,e)\}$. Both relations are (vacuously) transitive. Then $(a,c) \in R \circ S$ (since $aRb$ and $bSc$) and $(c,e) \in R \circ S$ (since $cRd$ and $dRe$) but $(a,e) \notin R \circ S$. Thus $R \circ S$ is not a transitive relation.

Answer 3

The transitive property must be true for every a,b,and c in X not just for some a,b,and c. For example the relation "likes" is not transitive because if Jeff likes Jerry and Jerry likes Pat, it does not necessarily implies that Jeff like Pat. However you may find three persons for which this relation holds. "For all" should not be replaced by "for some".