Which statement is not a mistake that Reina has made?

"A survey done at a certain high school found that any student who liked tennis also liked swimming. They also found that students only liked swimming if they could swim."

Reina: If 30 students from the high school can swim, then 30 students from the school also like tennis.

If the quoted paragraph above is true, which of the following is NOT a mistake that Reina has made?

• a. Some people may like tennis only because they like swimming
• b. Some people may not like swimming even though they can swim
• c. Some people may not like any sport but swimming
• d. Some people may not like tennis even though they can swim

I got option d as the answer, because if you told Reina about option d statement, she would disagree, hence that option does not fall under the mistake Reina has made/assumed.

Am I wrong to think this?

• The claim made by $d$ is true...nothing in the problem statement tells us that swimmers like tennis, only the other way round. So if $R$ disagrees with it, she is making a mistake.
– lulu
Commented Jul 13 at 10:18

Using the framework of propositional logic, write $$T$$ for "likes tennis", $$S$$ for "likes swimming" and $$C$$ for "can swim". The first paragraph says that $$T \to S$$ and $$S \to C$$.

Reina concludes from this that $$C \to T$$, which does not follow (we only have $$T \to C$$).

Then, the options are:

a. Not completely straightforward to interpret, but this basically says $$\neg(T \to C)$$

b. $$\neg(C \to S)$$

c. $$\neg(S \to T)$$

d. $$\neg(C \to T)$$

It is then clear that the only incorrect option is a, because we do have $$T \to C$$ while none of the other implications are assumed to hold.

"A survey done at a certain high school found that any student who liked tennis also liked swimming. They also found that students only liked swimming if they could swim."

Reina: If 30 students from the high school can swim, then 30 students from the school also like tennis.

If the quoted paragraph above is true, which of the following is NOT a mistake that Reina has made?

• a. Some people may like tennis only because they like swimming
• b. Some people may not like swimming even though they can swim
• c. Some people may not like any sport but swimming
• d. Some people may not like tennis even though they can swim

The quoted paragraph tells us that Tennis is a subset of Swim and that Swim is a subset of Canswim, that is, $$T\subseteq S\subseteq C.$$ Observe that this is consistent with all of statements a to d.

Now, Reina's claim together with the quoted paragraph means that $$T=S=C.$$ This agrees with option a, and disagrees with statements b-d.

Hence, statements b-d, but not statement a, are Reina's mistakes.

I got option d as the answer, because if you told Reina about option d statement, she would disagree, hence that option does not fall under the mistake Reina has made.

Statement d contradicts neither the quoted paragraph nor Reina's claim, but does contradict the conjunction of the quoted paragraph and Reina's claim. Therefore, Reina is mistaken about this statement.