The problem of choosing the correct statement.

Question

This is a problem of choosing the correct statement.

In a certain class of a school, we have investigated whether each student likes candy, pizza, chocolate, and egg. Then, from the result, we know

・ all students that like candy like egg

・ all students that like chocolate don't like egg

・ all students that don't like chocolate like pizza

・ there is no student who likes all four kinds of food.

Then, choose what we can say.

1. There is(are) a(some) student(s) who like(s) pizza and chocolate.

2. There is(are) a(some) student(s) who like(s) only chocolate, i.e., like chocolate but don't like other food.

3. There is no student who likes three kinds of food.

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I want to find the answer by using logic.

Let me regard $Ca$ as "like candy" and $\overline {Ca}$ as "don't like candy", and so on. (pizza : $P$, chocolate : $Ch$, egg : $E$)

Then, we know

・$Ca \implies E$

・$Ch \implies \overline E$

・$Ch\implies P$

・$\forall x\in [this\ class]$, $x\in (Ca)^c\cup P^c\cup (Ch)^c\cup E^c$

Now, since the class is not empty, there is some student $x.$ Then, from the fourth condition, $x\in (Ca)^c\cup P^c\cup (Ch)^c\cup E^c.$

・If $x\in (Ca)^c$, then I think there is nothing we can say from other conditions.

・If $x\in P^c$, then $x\in (Ch)^c$ from the third condition.

・If $x\in (Ch)^c$, then there is nothing to say.

・If $x\in E^c$, then $x\in (Ca)^c$.

I cannot proceed from here and cannot find the correct statement from 1-3. Is it difficult to solve this problem by using logic ?

I'd appreciate any idea.

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Add:

The answer is 2. However, if so, I think the additional condition is needed : [there is at least one student that likes chocolate ($Ch\neq\emptyset$).]

Answer 1

None of these can be logically inferred. The simplest way to see this is to make a model satisfying the conditions and contradicting a specific one of the answers. For $1$ and $2$ take a single student that likes pizza. For $3$ take a student which likes egg, candy and pizza. So none of these statements follow from the conditions.

Answer 2

As suggested in the Comments, a Venn Diagram is handy here. Let's start with drawing a Venn diagram for the 4 categories we are dealing with there: students who like candy, students who like egg, students who like chocolate, and students who like pizza:

Now we can shade areas where we know no students can exist. The red areas have to be empty because of claim 1 (we cannot have any students who like candy but don't like egg), the brown areas are due to claim 2( we cannot have any students who like chocolate and egg), the orange areas are due to claim 3 (we cannot have any students who don't like chocolate, and don't like pizza), and the green one is due to claim 4 (no students likes all 4):

OK, so this tells us there can only be 5 types of students:

1. Students who like only pizza
2. Students who like pizza and egg, but not chocolate or candy
3. Students who like pizza and chocolate, but not egg or candy
4. Students who like only chocolate
5. Students who like candy, egg, and pizza ... but not chocolate.

However, for each of these types , we don;t know if there is in fact any student like that. In fact, for all we know, there are no students at all!

This means that:

Claim 1 could be true, but we can not say that for certain. Maybe no students like chocolate, or maybe there are some students that like chocolate, but not pizza.

Claim 2 could be true, but we are again not certain of that. And note that knowing that there is at least one student who likes chocolate does not make claim 2 true either, because maybe there is only one such student, and that student also likes pizza.

Finally, claim 3 could be true: if there is no student that likes candy, egg, and pizza, then there are no students at all that like 3 foods. But again we are not certain of that: maybe there is a student that likes egg, candy, and pizza.

In sum, none of the 3 statements logically follow from the given statements, because they can all be false even if all the givens are true.