# How do I express “A person in this arcade won a grand prize” and “Nobody in this arcade played Pac-man”

If we have

A(X) = x is in this arcade

W(X) = "x won a grand prize"

P(x) = "x played Pac-Man"

how would I express "A person in this arcade won a grand prize," and "Nobody in this arcade played Pac-Man." As premises in a quantified logical statement.

What I have done so far is $$\forall x(\neg A(x) \implies P(x))$$ and $$\exists x (A(x) \implies W(x)$$

• I don't think that I'm allowed to answer until you've shown work. Place to start would be to consider the various ways of using the tools: $\forall x, ~\exists x,~$ and $~\neg (\cdots)$. From this perspective, what have you tried and where are you having trouble? – user2661923 Oct 3 '20 at 18:15
• yeah I edited the original post to show what I think the answer is – ej21vf Oct 3 '20 at 18:27

"A person in this arcade won a grand prize"

$$\exists x (A(x) \implies W(x))$$

See e.g. here for a detailed explanation of why your formulization is incorrect: As a rule of thumb, with $$\exists$$ use $$\land$$ and with $$\forall$$ use $$\implies$$, so instead write

$$\exists x(A(x) \land W(x))$$

"Nobody in this arcade played Pac-Man."

$$\forall x(\neg A(x) \implies P(x))$$

Your formula says "Everyone who is not in this arcade played Pac-Man". But what you want to say is "Everyone who is in this arcade did not play Pac-Man", that is, the negation should be applied to the playing Pac-Man, not to the being in the arcade:

$$\forall x (A(x) \implies \neg P(x))$$

• +1 nice detail.... – user2661923 Oct 4 '20 at 9:57

$$[~\exists x ~(A(x) \wedge W(x))~]$$
and
$$\{\neg ~~[\exists x ~(A(x) \wedge P(x))~]~~\}$$.