Express the quantified statement in an equivalent way.
Some students are bored.
Would it be Not all students are bored?
How about
Not all students are not bored.
no try this ALL STUDENTS ARE BORED
No, "Not all students are bored" is not an equivalent expression to "Some students are bored." The correct equivalent expression would be "There exists at least one student who is bored."
To understand why, let's break it down:
The original statement "Some students are bored" can be translated into logical notation as ∃x (S(x) ∧ B(x)), where S(x) represents "x is a student" and B(x) represents "x is bored." The existential quantifier (∃) indicates that there exists at least one value of x that satisfies the conditions S(x) and B(x).
On the other hand, the expression "Not all students are bored" can be translated into logical notation as ¬∀x (S(x) → B(x)). This expression uses the universal quantifier (∀), which means "for all" or "every." The negation symbol (¬) negates the entire expression "∀x (S(x) → B(x))."
Essentially, ¬∀x (S(x) → B(x)) says that it is not the case that every student being bored. In other words, there could be some students who are not bored, but it doesn't guarantee that there are any bored students.
Therefore, "Not all students are bored" does not fully capture the meaning of "Some students are bored."