Using the statement “at least” in propositional logic

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?

$(1)\;\text{ should be }\;s\vee e\vee m$

$$s \lor e \lor m\;\text{ asserts that at least one (or perhaps two, maybe even all three) of }\;s, \,e,\, m \;\text{ holds.}$$ $(2)$ 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: $\;\;\lnot(m \land s)\rightarrow \lnot e \equiv (m\land s) \lor \lnot e$

1. $s\vee e\vee m$
2. $m\rightarrow (\neg s\wedge\neg e)$
3. $\neg(m\wedge s)\rightarrow\neg e$
• You should add parentheses to make statement 2 and 3 less ambiguous. – MM8 Feb 17 '17 at 17:20
• You are right, thanks for point it out – John Feb 17 '17 at 17:21
• Statement three should be $$\lnot (m \land s)\rightarrow \lnot e \equiv (\lnot m \lor \lnot s)\rightarrow \lnot e$$ – Namaste Feb 17 '17 at 17:24
• Agreed, "If you don't use magic mushrooms and spider legs" is negating the conjunction, not the individual statements. – MM8 Feb 17 '17 at 17:25
• Well, @Timon, after our comments, John corrected his post, but never acknowledged the error. – Namaste Feb 17 '17 at 17:27