Question

Answer the following questions: % Question (i) Write the negation of the statement: \( \exists n \in \mathbb{N} \) such that \( n + 8 > 11 \).

Hint:

To negate an existential quantifier (\( \exists \)), change it to a universal quantifier (\( \forall \)) and negate the condition.

Answer 1

Step 1: Identify the original statement.
The original statement is: \[ \exists n \in \mathbb{N} \text{ such that } n + 8 > 11 \] This means that there exists a natural number \(n\) such that \(n + 8 > 11\), or equivalently, \(n > 3\).

Step 2: Negate the statement.
The negation of an existential statement (\( \exists \)) becomes a universal statement (\( \forall \)). Thus, the negation of the original statement is: \[ \forall n \in \mathbb{N}, \; n + 8 \leq 11 \] Or equivalently, \[ \forall n \in \mathbb{N}, \; n \leq 3 \]

Final Answer: \[ \forall n \in \mathbb{N}, \; n \leq 3 \]