Using inference to prove that $(p\wedge q)\vee (p\wedge r) \to (q \vee r)$.

Question

Only using rules of inferences to prove that $(p\wedge q)\vee (p\wedge r) \to (q \vee r)$

I tried to solve it with this:

1. (p⋀q)⋁(p⋀r) (premise)
2. (p⋀q) (Assumption)
3. p (⋀-elimination)
4. p⋀r (Assumption)
5. p (⋀-elimination)
6. p (⋁-elimination 3,5)

But this seems wrong. Does anyone else have a clue on which rules to apply for this? Thank you.

Answer 1

$\def\fitch#1#2{\quad\begin{array}{|l}#1\\\hline #2\end{array}}$

I tried to solve it with this:

1. (p⋀q)⋁(p⋀r) → (q⋁r) (premise)
2. (p⋀q) (Assumption)
3. p (⋀-elimination)
4. p⋀r (Assumption)
5. p (⋀-elimination)

Don't ever start by premising what you seek to prove.   This is a proof with no premises.

Since you want to introduce a conditional, use a conditional proof -- assume the antecedant in order to derive the consequent.

Since that assumption is a disjunction, eliminate it with a proof by cases. In each case assume the left or right of the disjunction and aim to derive the consequent.

Since each case is a conjunction, eliminate it to the part that enables you to derive the consequent by introducing a disjunction.

Finish by introducing the conditional and you are done.

Fill in the dots.

$$\fitch{}{\fitch{\ldots}{\fitch{\ldots}{\ldots\\\ldots}\\\fitch{\ldots}{\ldots\\\ldots}\\\ldots}\\((p\land q)\lor(p\land r))\to(q\lor r)}$$