Truth table for $(p \implies q) \lor (q \implies r) \lor (r \implies p)$ : What should my next step be?

Question

I am working on a truth table for $(p \implies q) \lor (q \implies r) \lor (r \implies p)$

This is what I have done so far:

• My next step would be to do the disjunction from the first two implications
• I'm not too sure about the final step, would I just do the second disjunction from the second and third implications?

Answer 1

There's nothing more to be said than what Callus remarked:

The $\lor$ operator is associative, so you can do whichever one you want first. But, let's say you do the disjunction of the first two implications. Then the last step is the disjunction between THAT disjunction and the third implication.