Are these two statements logically equivalent? XOR and If/then Consider the statement "Everyone in Florida plays either basketball or soccer, but no one plays both basketball and soccer." Let $B(x)$ be the propositional function "$x$ plays basketball", and $S(x)$ be "$x$ plays soccer".
I am expressing this statement using quantifiers, and am curious if the following are logically equivalent.


*

*$\forall x (B(x) \oplus S(x)) $


logically equivalent to 


*$\forall x ((B(x) \implies \neg S(x)) \land (S(x) \implies \neg B(x))) $


No. The second formula can be satisfied by a population of couch potatoes ie $\neg B(x) \land \neg S(x)$ is consistent with the second formula, but not the first.
