Question

Which of the following statements is NOT true? (The names of the predicates are intuitive.)

• A. \(\forall x\, \forall y\; Classmate(x,y) \Rightarrow Classmate(y,x)\)
• B. \(\forall x\; Likes(x, Icecream) \Rightarrow \neg \exists x\; \neg Likes(x, Icecream)\)
• C. "Each king is a person" is equivalent to \(\forall x\; IsKing(x) \wedge IsPerson(x)\)
• D. "All humans are mortal" is equivalent to \(\forall x\; IsHuman(x) \Rightarrow IsMortal(x)\)

Hint:

"Every A is a B" translates using implication (\(A(x) \Rightarrow B(x)\)), never a conjunction (\(A(x) \wedge B(x)\)).

Answer 1

The Correct Option is C

This question is really about one recurring mistake in first-order logic translation: mixing up $\wedge$ (and) with $\Rightarrow$ (implies) when turning a universal English sentence into a formula. Let's walk through all four options.

1. Classmate symmetry: $\forall x \forall y\; Classmate(x,y) \Rightarrow Classmate(y,x)$. Being classmates is a two-way relationship by definition, if x shares a class with y, y shares that same class with x. This formula states exactly that symmetry correctly, so it is true.
2. The ice cream tautology: $\forall x\; Likes(x, Icecream) \Rightarrow \neg\exists x\, \neg Likes(x, Icecream)$. Notice that $\neg\exists x\, \neg Likes(x, Icecream)$ is just another way of writing $\forall x\; Likes(x, Icecream)$ (saying "there is no one who dislikes it" is the same as saying "everyone likes it"). So this statement has the shape $P \Rightarrow P$, which is always true regardless of the actual world, a logical tautology. True.
3. The king-person conjunction: the English says "Each king is a person", which is a conditional claim, it only talks about objects that ARE kings. The correct FOL form is $\forall x\; IsKing(x) \Rightarrow IsPerson(x)$. But the option instead writes $\forall x\; IsKing(x) \wedge IsPerson(x)$, using AND. That formula says literally every object in the universe, whether it is a king or not, must be both a king and a person, which is a far stronger and generally false claim, and it does not match the original English sentence at all. This equivalence claim is broken, so this statement is NOT true, this is our answer.
4. Humans and mortality: $\forall x\; IsHuman(x) \Rightarrow IsMortal(x)$ is the textbook correct translation of "All humans are mortal", using the right implication structure. True.

Three out of four options hold up under inspection, only the king-person statement swaps the implication for a conjunction, breaking the claimed equivalence. So the answer is option (C).

\[ \boxed{\text{Option (C)}} \]