logic connectives and symbols "most"

Question

"She went to at most two of the cities"

anwser: ¬(a∧b∧c) or ¬a∨¬b∨¬c

I don't really understand the logic here... For me, the answer means she went to none of those cities

Answer 1

Assume that $a$ means "she went to city $A$" and so on.

Thus $¬(a∧b∧c)$ means "it is not true that she went at $A$ and $B$ and $C$".

This formula is TRUE when at least one of $a, b, c$ is FALSE.

And this is consistent with the fact that "She went to at most two of the cities": this means that she did not go to all three cities.

The part $¬a ∨ ¬b ∨ ¬c$ means "either she did not go to $A$ or she did not go to $B$ or she did not go to $C$.

This formula is FALSE when $a,b,c$ are all TRUE.

And this is consistent with the fact that "She went to at most two of the cities": the statement that she went to all three must be false.

Answer 2

I'm not sure if you are confusing the symbols for "and" and "or." $\vee$ means "or" and $\wedge$ means "and." Therefore, $\neg(a\wedge b\wedge c)$ means "it is not the case that she went to all the cities." Similarly, $\neg a \vee\neg b\vee\neg c$ means "there is at least one city she didn't visit."

Answer 3

Re: your interpretation of the answer, I think you're switching "$\vee$" ("or") and "$\wedge$" ("and"). "She went to none of those cities" would be "$\neg a\wedge \neg b\wedge \neg c$."

As to the answer itself, "She went to at most two of the cities" is the same as "There was (at least) once city she didn't go to." That's why "$\neg a\vee\neg b\vee\neg c$" is the right answer: it's saying that she missed at least one city, which - since there are only three cities - is the same as saying that she went to at most two cities.