So I am working on logical equivalences for the first time and it was all making sense, until I was given the exercise:
Verify the following equivalence by writing an equivalence proof. That is, start on one side and use known equivalences to get to the other side. $(p \to q) \land (p \lor q) ≡ q$.
I am aware of the various laws, however in what order do I apply these laws? Is there an order? If there is not an order then what are the steps to take in order to prove the equivalence?
\begin{align} (p \to q) \land (p \lor q) &\equiv (\lnot p \lor q) \land (p \lor q) &&\text{classical definition of }\to \\ &\equiv (\lnot p \land p) \lor q &&\text{distributivity of } \lor \text{ over } \land \\ &\equiv \bot \lor q \equiv q \end{align} where $\bot$ is the absurdity, i.e. a proposition that is always false.