Knowing one term is false, how can I prove the other is true in a disjuntion?

Question

How can I prove this true:

$$p \vee q, \neg q \vdash q$$

Using natural deduction?

I have tried to eliminate the disjunction but didn't go anywhere and working with proof by contradiction didn't help me either..

Answer 1

Using the following proof tree:

$\cfrac{\cfrac{p\vee q\quad \cfrac{{}^\dagger p^1 \quad \neg p}{\bot}\,\neg E \quad \cfrac{{}^\dagger q^2 \quad {}^\dagger (\neg q)^3}{\bot} \, \neg E}{\bot} \,\vee E,1,2}{q} \,\bot E,3$

${}^\dagger$ denotes marked assumptions, $\neg E$ is negation elimination, $\vee E$ is or elimination and $\bot E$ is absurdity elimination. Notation is from Sets, Models and Proofs