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.
On
$$\begin{align} p ⇒ (q∨r) ≡ (p∧(\neg r)) \rightarrow q &\iff \lnot p \lor (q \lor r) \equiv \lnot (p \land \lnot r) \lor q \\ & \iff \lnot p \lor q \lor r \equiv (\lnot p \lor r) \lor q \\ & \iff \lnot p \lor q \lor r \equiv \lnot p \lor r \lor q \\ & \iff \lnot p \lor q \lor r \equiv \lnot p \lor q \lor r\end{align}$$
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.
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.