$p ⇒ (q∨r) ≡ (p∧(\neg r)) ⇒ q$ are logical equivalent? I have determine whether the following equivalence is true or not $$p ⇒ (q∨r) ≡ (p∧(\neg r)) ⇒ q$$ using logical equivalences definitions.
I am never able to do these sorts of questions correctly no matter what. It is just really hard for me to do these. I was really hoping someone could help me out with this since it seems no matter what I do for this question it is wrong. 
 A: When all else fails, you can always write the truth table for every expression and see if they are the same.
In this case, I suggest you first transform the two $A\implies B$ statements into $\neg A \vee B$, then continue from there on. Remember, you can always double check each step with truth tables! For example, if you have expression $A$ which you change into $A_1$ and then into $A_2$, you can write down all three tables and see if they are the same.
A: $$\begin{align} p ⇒ (q∨r) ≡ (p∧(\neg r)) \rightarrow q &\iff \lnot p \lor (q \lor r) \equiv \lnot (p \land \lnot r) \lor q \\ 
& \iff \lnot p \lor q \lor r \equiv (\lnot p \lor r) \lor q \\ 
& \iff \lnot p \lor q \lor r \equiv \lnot p \lor r \lor q \\
& \iff \lnot p \lor q \lor r \equiv \lnot p \lor q \lor r\end{align}$$
You'll see that that indeed, the right hand side of the $\iff$ connective is true for all p, q, r, and hence so must be the left-hand side.
Remark: Note that the strategy above was to modify (into an equivalent statement) each side of the equivalence in question: $p ⇒ (q∨r) ≡ (p∧(\neg r)) $, until they became identical, thereby proving equivalence is valid. (Alternatively, if it had proved not to be a valid equivalence, working this way can be helpful to arrive at a proposed equivalence that is more readily seen to be invalid.)  
In the case of valid equivalences: This approach can be seen as working from the left side to the right-side, and the right-side to the left-side, until they "meet in the "middle." A proof, then, which manipulates only the left-side of an equivalence can easily be constructed to arrive, validly, at the right-hand side of an equivalence.
Look closely at each line to try to find the justification for each line.
