How I express these sentences in logic?

Question

$A: $ "Amelia goes to birthday party"

$B: $ "Bart goes to birthday party"

$C: $ "Chris goes to birthday party"

$D: $ "Diana goes to birthday party"

Let's say I want write this in logic language: "If Amelia and Bart go to birthday party, then Chris or Diana also go to birthday party".

I do it like this: $(A \wedge B) \rightarrow (C \vee D)$

Now If I need to write this in logic: "Bart may never go alone to birthday party." I'm confused because I don't know how to interpret this correctly. I have thought about simply writing: $\neg B$, but this would mean "Bart does not go to birthday party".

Then I thought I could interpret it that way better: "If Bart goes to birthday party, then at least someone else goes to birthday party too".

This in logic should be:

$$B \rightarrow ((A\wedge C \wedge D) \vee (A \wedge C) \vee (A \wedge D) \vee (C \wedge D) \vee (A) \vee (C) \vee (D))$$

Is it fine like that or completely wrong? I have thought about this for a while myself but I'm not sure! :c

Answer 1

Its correct but it would be simpler if you wrote: $B \rightarrow (A\vee C\vee D)$. Since this also encompasses the other cases that you have written in your formula.

Answer 2

One way to say it would be the statement. "It is not the case that B goes to the party and none of A, or C or D go to the party".

$\neg (B\wedge(\neg A\wedge\neg C\wedge \neg D))$

Or it could be written as "If B goes to the party, A or C or D go to the party" This could be written as $B\implies (A\vee C\vee D)$

A third way is to say "if B goes, then it is not the case that none of A or C or D go" This is expressed as $B\implies (\neg(\neg A\wedge\neg C\wedge \neg D))$