Questions: Negation of a quantified statement Consider the given statement. None of the rooms at the hotel have a bed. For each statement below, determine whether it is a negation of the given statement. - All rooms at the hotel have a bed. No - Not all rooms at the hotel have a bed. Yes No - Some rooms at the hotel have a bed. Yes No - Some rooms at the hotel do not have a bed. Yes No

Negation of a quantified statement

Consider the given statement. None of the rooms at the hotel have a bed. For each statement below, determine whether it is a negation of the given statement.

- All rooms at the hotel have a bed. No
- Not all rooms at the hotel have a bed. Yes No
- Some rooms at the hotel have a bed. Yes No
- Some rooms at the hotel do not have a bed. Yes No

Transcript text: Negation of a quantified statement Consider the given statement. None of the rooms at the hotel have a bed. For each statement below, determine whether it is a negation of the given statement. \begin{tabular}{|l|c|c|} \hline & Yes & No \\ \hline All rooms at the hotel have a bed. & 0 & $\bigcirc$ \\ \hline Not all rooms at the hotel have a bed. & $\bigcirc$ & $\bigcirc$ \\ \hline Some rooms at the hotel have a bed. & $\bigcirc$ & $\bigcirc$ \\ \hline Some rooms at the hotel do not have a bed. & $\bigcirc$ & $\bigcirc$ \\ \hline \end{tabular}

Solution

Solution Steps

To determine the negation of the given statement "None of the rooms at the hotel have a bed," we need to find a statement that contradicts it. The original statement implies that there is not a single room with a bed. Therefore, the negation would be that there is at least one room with a bed. We will evaluate each provided statement to see if it matches this negation.

Step 1: Identify the Original Statement

The original statement is $ \text{None of the rooms at the hotel have a bed.} $. This can be expressed mathematically as $ \forall x \, (R(x) \rightarrow \neg B(x)) $, where $ R(x) $ indicates that $ x $ is a room and $ B(x) $ indicates that room $ x $ has a bed.

Step 2: Determine the Negation

The negation of the original statement is that there exists at least one room that has a bed. This can be expressed as $ \exists x \, (R(x) \land B(x)) $, meaning "Some rooms at the hotel have a bed."

Step 3: Evaluate Each Statement

We evaluate the provided statements against the negation:

$ \text{All rooms at the hotel have a bed.} $ - This contradicts the negation, so it is not a negation.
$ \text{Not all rooms at the hotel have a bed.} $ - This does not directly assert that some rooms have beds, so it is not a negation.
$ \text{Some rooms at the hotel have a bed.} $ - This directly matches the negation, so it is a negation.
$ \text{Some rooms at the hotel do not have a bed.} $ - This does not contradict the original statement but does not assert the existence of rooms with beds, so it is not a negation.