Question

Which of the following is the negation of the statement "for all M>0, there exists x$\in$S such that x $\ge$ M" ?

• A. there exists M>0, there exists x$\in$S such that x<M
• B. there exists M>0, there exists x$\in$S such that x $\ge$ M
• C. there exists M>0, such that x<M for all x$\in$S
• D. there exists M>0, such that x $\ge$ M for all x$\in$S

Hint:

To negate a quantified statement, systematically switch every 'for all' ($\forall$) to 'there exists' ($\exists$) and vice versa, and then negate the final condition. For example, the negation of "All cats are black" is "There exists a cat that is not black".

Answer 1

The Correct Option is C

Let's break down the original statement using quantifiers.
The statement is: $\forall M>0, \exists x \in S, x \ge M$.
In words, this says "For every positive number M, we can find an element x in the set S that is greater than or equal to M." This is the definition of the set S being unbounded above.
To negate a statement with quantifiers, we follow these rules:
1. Change the universal quantifier ($\forall$, "for all") to an existential quantifier ($\exists$, "there exists").
2. Change the existential quantifier ($\exists$) to a universal quantifier ($\forall$).
3. Negate the predicate (the condition at the end).
Let's apply these rules step-by-step to the statement $P: \forall M>0, \exists x \in S, x \ge M$.
The negation, $\neg P$, will be:
1. Change $\forall M>0$ to $\exists M>0$.
2. Change $\exists x \in S$ to $\forall x \in S$.
3. Negate the predicate $x \ge M$. The negation of $x \ge M$ is $x<M$.
Combining these, the negated statement is:
$\exists M>0, \forall x \in S, x<M$.
In words, this says "There exists a positive number M, such that for all elements x in the set S, x is less than M." This is the definition of the set S being bounded above.
This matches option (C).