I am studying for my exams and I am facing some issues with the "at least" statement in propositional logic.
Mary wants to mix a magic potion. Here is the recipe.
(1): You need at least one of the following ingredients: spider legs, eyes of a toad, magic mushrooms.
(2): If you are using magic mushrooms, then you can not use the other two ingredients.
(3): If you don't use magic mushrooms and spider legs, then you aren't allowed to use eyes of a toad.
Let
- $s =$ spider legs
- $m =$ magic mushrooms
- $e =$ eyes of a toad
My solution would be something like this, but I am not sure about the first one. Feedback would be nice
- (1) $= (m \vee e) \wedge (m \vee s) \wedge (e \vee s)$
- (2) $= (\neg s \wedge \neg e) \rightarrow m$
- (3) $= (\neg m \wedge \neg s) \rightarrow (\neg e)$
Is the first equation correct?
Best Answer
$(1)\;\text{ should be }\;s\vee e\vee m$
$$s \lor e \lor m\;\text{ asserts that at least one (or perhaps two, maybe even all three) of }\;s, \,e,\, m \;\text{ holds.}$$ $(2)$ We have $\quad m\rightarrow \lnot( s \lor e) \equiv m\rightarrow (\neg s\wedge\neg e)$
$\quad$ This means that "If we use magic mushrooms, then we cannot use (spider legs or eyes of a toad)", or equivalently, "If we use magic mushrooms, then we cannot use spider legs, and we cannot use eyes of a toad."
$(3)$ We can express this statement into propositional logic notation: $\;\;\lnot(m \land s)\rightarrow \lnot e \equiv (m\land s) \lor \lnot e$