I came across the following problem in a book:
Show that if $p, q$, and $r$ are compound propositions such that $p$ and $q$ are logically equivalent and $q$ and $r$ are logically equivalent, then $p$ and $r$ are logically equivalent.
The book's solution certainly makes sense:
To say that $p$ and $q$ are logically equivalent is to say that the truth tables for $p$ and $q$ are identical; similarly, to say that $q$ and $r$ are logically equivalent is to say that the truth tables for $q$ and $r$ are identical. Clearly if the truth tables for $p$ and $q$ are identical, and the truth tables for $q$ and $r$ are identical, then the truth tables for $p$ and $r$ are identical. Therefore $p$ and $r$ are logically equivalent.
I decided to "symbolically translate" the problem in the book:
Show that $[(p\leftrightarrow q)\land(q\leftrightarrow r)]\to(p\leftrightarrow r)$ is a tautology.
I wrote out a truth table and everything checks out, as expected (and as mentioned in the book's solution). My question is whether or not there is a more "algebraic" solution using equivalences (not resorting to CNF or DNF).
Any ideas?
Best Answer
I managed to work out a solution, and I thought I would share even though it is rather hideous (picture posted to make formatting much more pleasant to read).