Prove a must be true if b is not for premise either a or b but not both!

Question

I have $(A \wedge \neg B) \vee (\neg A \wedge B)$ as a premise and I've concluded that $\neg B$ is true, therefore A must be true, but how would I do this using natural deduction and the rules I can use?

Answer 1

What you are trying to derive is a conditional, so conditional introduction will be used: it is a conditional proof, assume the antecedent aiming to derive the consequent.

As Mauro ALLEGRANZA commented, your premise is a disjunction, so disjunction elimination will be involved. That is a proof by cases.

Next, both the disjuncts in the premise are conjunctions, so conjunction elimination is indicated. Clearly that will immediately derive the required consequent in one case. I'll leave doing so in the other to you. $\def\fitch#1#2{\quad\begin{array}{|l}#1\\\hline#2\end{array}}$

$$\fitch{(A\wedge\neg B)\vee(\neg A\wedge B)}{\fitch{\neg B\qquad\textsf{Assumption}}{\fitch{A\wedge\neg B\qquad\textsf{Assumption}}{A\qquad\textsf{Conjunction Elimination}}\\\fitch{\neg A\wedge B\qquad\textsf{Assumption}}{~\vdots\qquad\textsf{Conjunction Elimination}\\~\vdots\qquad\ldots\\A\qquad\ldots}\\A\qquad\textsf{Disjunction Elimination}}\\\neg B\to A\qquad\textsf{Conditional Introduction}}$$