Which of the following NOT true? (The name of the predicate are intuitive)

The correct option that is NOT true is:
"Each King is a person" is equivalent to ∀ Isking (x) ∧ Isperson(x)

Step-by-Step Explanation:

1. Analyzing the translation of "Each King is a person":
The statement "Each King is a person" (or "Every king is a person") means that if any object in the domain is a king, then that object must also be a person. It does not assert that everything in the universe is a king, only that being a king implies being a person.
In first-order logic, this universal conditional statement is represented using a universal quantifier and an implication (⇒) operator:

$\forall x\left(\mathrm{Isking}\right(x)\Rightarrow \mathrm{Isperson}(x\left)\right)$

2. Why the conjunction (∧) translation is incorrect:
The option states that the sentence is equivalent to a formula using a conjunction (∧):

$\forall x\left(\mathrm{Isking}\right(x)\wedge \mathrm{Isperson}(x\left)\right)$

This formula translates to: "Everything in the universe is a king AND everything in the universe is a person." This is incorrect because it requires every single object in the domain to be both a king and a person simultaneously. Since there are many objects in the world that are not kings, this translation is incorrect, making the option's claim false.

3. Verification of other options (which are TRUE):
• ∀x likes (x, Ice-cream) ⇒ ¬∃x¬ likes (x, Ice-cream): This is a true statement. If everyone likes ice cream, then it is not the case that there exists someone who does not like ice cream.
• ∀x ∀y classmate (x, y) ⇒ Classmate (y, x): This is true because the "classmate" relation is symmetric by nature.
• "All humans are mortal" is equivalent to ∀x is human (x) ⇒ Ismortal(x): This is the correct standard translation of a universal conditional statement using implication.