Intuitive explanation for p ∨ q → r ≡ ( p → r) ∧ (q → r) Although, it is possible to prove the above equivalence using truth tables, I don't know how to prove it without using truth tables.Can someone explain it in plain english?
 A: This is sometimes referred to as Inference by Cases.
It's easy if you bear in mind that $\mathcal{S} \Rightarrow \mathcal{T}$ is equivalent to $(\lnot \mathcal{S}) \lor \mathcal{T}$ where $\mathcal{S}$ and $\mathcal{T}$ are any wffs.
\begin{align}
[(P \lor Q) \Rightarrow R] &\Leftrightarrow [\lnot(P \lor Q) \lor R] \\
&\Leftrightarrow [(\lnot P \land \lnot Q) \lor R] \quad \text{DeMorgan's} \\
&\Leftrightarrow [(\lnot P \lor R) \land (\lnot Q \lor R)] \\
&\Leftrightarrow [(P \Rightarrow R) \land (Q \Rightarrow R)].
\end{align}
A: This is a neat little identity. I would put it like this: there's a red button that sets off the bomb. If either Angie or Bill pushes it, then the bomb will go off.
$$(\text{Angie pushes the button}) \vee (\text{Bill pushes the button}) \implies (\text{explosion})$$
But we also know that, independently, if Angie pushes the button then the bomb will go off and if Bill pushes the button, the bomb will go off.
$$
((\text{Angie pushes the button}) \implies (\text{explosion}))
\wedge
((\text{Bill pushes the button} \implies (\text{explosion}))$$
A: Perhaps it's best to think about the only way an implication can fail: The hypothesis must be true and the conclusion false. So the left statement fails if and only if $P\vee Q$ is true and $R$ is false, i.e., if at least one of $P$ and $Q$ is true and $R$ is false. The right statement fails if and only if (at least) one of $P\implies R$ and $Q\implies R$ is false, so either $P$ or $Q$ must both be true and $R$ must be false.
A: It helps to think of examples. Let $p$ be "Alfred wants to play basketball," $q$ be "Bertrand wants to play basketball," and $r$ be "We will play basketball."
The first statement reads "If either Alfred or Bertrand want to play basketball, we will play basketball." The second statement reads "If Alfred wants to play basketball we will play basketball; if Bertrand wants to play basketball we will play basketball." Do you see why these are the same?
This isn't a formal proof, but you asked for an intuitive explanation.
