How can you show that $(p \rightarrow q) \land (q \rightarrow r) \leftrightarrow (p \rightarrow r)$ is not a tautology?

Question

Obviously, $(p \rightarrow q) \land (q \rightarrow r) \rightarrow (p \rightarrow r)$ due to Hypothetical Syllogism, but how about the converse? How can I disprove the statement?

Answer 1

It is not a tautology; so you cannot prove it.

Consider $p$ and $r$ as true but $q$ as false. Thence $p\to r$ is true, but $p\to q$ is false (and hence $(p\to q)\land(q\to r)$ too).