This has been asked before, but I have different problems. I’m asking because this was not discussed in class and I’m unsure of the procedure in obtaining the proof. The two in question are the following (simple, I know, but I’m having serious trouble.)
A. $(p \land q) \to q$
B. $p \to (p \lor q)$
I feel really, really bad about not being able to understand these. According to the book, the answer is that A is a tautology by conjunction and that B is a tautology by disjunction. I understand that. However, I don’t think that my teacher will accept those as part of the proof — we have to use the laws (Idempotence, De Morgan’s, etc.).
Best Answer
Use the fact that $x \to y$ is equivalent to $\neg x \lor y$:
\begin{align*} (p \land q) \to q &\qquad\equiv\qquad \neg(p \land q) \lor q &\qquad\text{by definition of }\to \\ &\qquad\equiv\qquad \neg p \lor \neg q \lor q &\qquad\text{by DeMorgan's Law} \\ &\qquad\equiv\qquad \neg p \lor \top &\qquad\text{by Inverse Law} \\ &\qquad\equiv\qquad \top &\qquad\text{by Domination Law} \\ \end{align*}
You might have different names or use different notation here, but the same logic applies. See if you can do something similar for the other problem.