# Prove propositional formulas using Natural Deduction

(e) Show that $$\vdash \lnot(p \lor \lnot p) \to p \land \lnot p$$

(f) Show that $$\models p \lor \lnot p$$ and $$\vdash p \lor \lnot p$$. For the second part, you can assume (e), i.e. you can treat $$\lnot (p \lor \lnot p) \to p \land \lnot p$$ as an assumption.

The first part is natural deduction, I just can't figure out how to go about it, and the second part I'm really not sure about.

Thank you

$$% I'll add a link because I'm not sure how to put the symbols. [https://imgur.com/a/ijaE1vQ][1] [1]: https://imgur.com/a/ijaE1vQ$$

Assume $$\lnot (p \lor \lnot p)$$ and assume $$p$$. From it derive a contradiction and conclude with $$\lnot p$$ discharging the assumption.
Then assume $$\lnot p$$ and repeat, using double negation to conclude with $$p$$.
For $$\vDash p \lor \lnot p$$ use truth table.