Symbolising the set of pupils who do not like both subjects

Question

If $A= \{\text{pupils who like Science}\}$, $B= \{\text{pupils who like History}\},$ then is the set of pupils who do not like both subjects $$(A\cap B)^\complement$$ or $$(A\cup B)^\complement\,?$$

Answer 1

The phrase is ambiguous. It could have either of the two suggested meanings.

Answer 2

If $A= \{\text{pupils who like Science}\}$, $B= \{\text{pupils who like History}\},$ then is the set of pupils who do not like both subjects $$(A\cap B)^\complement$$ or $$(A\cup B)^\complement\,?$$

1. $(A\cap B)$ is the set of pupils who like both subjects, so its complement $\boldsymbol{(A\cap B)^\complement}$ is the set of pupils who like at most one subject.

2. The phrase ‘neither A nor B’ literally means ‘not (either A nor B)’, so $\boldsymbol{(A\cup B)^\complement}$ is the set of pupils who likes neither subject, which is precisely the set $(A^\complement\cap B^\complement)$ of pupils who dislike both subjects.

Your phrasing ‘not like both’ ambiguously could mean either of the above: is the negation ‘not’ being applied

1. to the phrase ‘like both’  (so, not (like both))

or

2. to the word ‘like’  (so, dislike both) ?

This is analogous to the difference between these sentences:

1. $¬∀x\, Lx$
2. $∀x\, ¬Lx\quad (\equiv ¬∃x\,Lx).$