# Is confidence transitive in asociation rules?

Given a set of rule such as:
$A \rightarrow B$; $B \rightarrow C$; that satisfy minimum confidence in the context of apriori algorithm meaning: $$\text{Conf}(A \rightarrow B) \geq \text{min. confidence}$$ and $$\text{Conf}(B \rightarrow C) \geq \text{min. confidence}$$

Is it possible to state that $\text{Conf}(A \rightarrow C) \geq \text{min. confidence}$?

I believe it's not possible to assert the later given that the relation between $\text{Support}(A \rightarrow B)$ and $\text{Support}(B \rightarrow C)$ doesn't give any information about the $\text{Support}(A \rightarrow C)$

Does this make sense?

Note: it's just a book exercise I'm trying to solve(not homework!) Here is the chapter link, it's in page 405 exercise 3.

$$\newcommand{\Conf}{{\rm Conf}} \newcommand{\confidence}{{\rm confidence}} \newcommand{\Support}{{\rm Support}}$$You cannot directly state that $$\Conf(A \rightarrow C) \geq \min \confidence$$. Because it depends on the support of $$A$$, $$B$$ and $$C$$. Let's say: \begin{align} \Support(A,B) &= 60\%, &\Support(A) &= 90\% \\ \Support(A,C) &= 20\%, &\Support(B) &= 70\% \\ \Support(B,C) &= 50\%, &\Support(C) &= 60\% \end{align} Let $$\min \confidence = 50\%$$, therefore: \begin{align} \Conf(A \rightarrow B) &= 66\% > \min \confidence \\ \Conf(B \rightarrow C) &= 71\% > \min \confidence \\ &{\rm But,} \\ \Conf(A \rightarrow C) &= 22\% < \min \confidence \\ \end{align}