TITLE: How do I prove the transitivity of a set of implications?
QUESTION [7 upvotes]: If I have a set of implications, how can I prove the transitivity? In other words: I know the transitivity law, but I need to show on paper for an assignment whether the argument is valid or not.
$$
\begin{align}
P&\to Q\\
Q&\to R \\
\therefore P&\to R
\end{align}
$$
I recall something to do with assuming P and/or Q, since $P\to Q$ will always be true if P is false anyhow, and the same with $Q\to R$... but I don't know how to do it on paper.

REPLY [5 votes]: Denote by $+$ the $or$ logic operator and by $\cdot$ the $and$ logic operator.
We have $(A\Rightarrow B)=\bar A+B$ and $(B\Rightarrow C) =\bar B+C$
Assume $(\bar A + B)=1=(\bar B+C)$.
Then
 $1=1\cdot 1=(\bar A + B)\cdot (\bar B+C) \\
 = (\bar A\bar B + \bar A C +  B\bar B + BC)\\
 = \bar A + BC
 = (\bar A + B) \cdot (\bar A+C)
 \\
 =\bar A + C
 $