Need help, new to truth tables and laws of logic (reading through Discrete Mathematics currently).
What I need is to show that $(p → q) → (p ∧ r) ≡ p ∧ (q → r)$ by using Laws of Logic. Any help would be great–I cannot find the relevant section in my book and I am unsure of the best approach to this kind of problem.
Best Answer
If you are working through Discrete Mathematics (which text?), then surely whatever section concerns logic is where you will find laws related to what you are trying to prove here. I will give a proof below and list the laws in the margin, but it will be completely useless unless you verify it through your own book (if you do not understand the proof, then it obviously does you no good at all). With that in mind, see if you can follow the proof outline below: \begin{align} (p\to q)\to(p\land r)&\equiv \neg(\neg p\lor q)\lor(p\land r)\tag{material implication}\\[1em] &\equiv (p\land\neg q)\lor(p\land r)\tag{De Morgan}\\[1em] &\equiv p\land(\neg q\lor r)\tag{distributivity}\\[1em] &\equiv p\land(q\to r).\tag{material implication} \end{align}