Question

quiz unit 3 - logic and geometry
14 of 34
this quiz: 34 point(s) possible
this question: 1 point(s) possible
use de morgans laws to write an equivalent statement for the sentence.
the hotel does not have a weight room or the conference center does not have an auditorium.
which statement is equivalent to the given sentence?
a. the hotel has a weight room and the conference center has an auditorium.
b. it is false that the hotel has a weight room or the conference center has an auditorium.
c. the hotel has a weight room and the conference center does not have an auditorium.
d. it is false that the hotel has a weight room and the conference center has an auditorium.

Explanation:

Step1: Recall De Morgan's Laws

De Morgan's Laws for logic state that:

• $

eg (p \land q) \equiv
eg p \lor
eg q$

• $

eg (p \lor q) \equiv
eg p \land
eg q$

Let $p$ be "The hotel has a weight room" and $q$ be "The conference center has an auditorium". Then the given statement is $
eg p \lor
eg q$.

Step2: Apply De Morgan's Law

Using the second De Morgan's Law $
eg (p \lor q) \equiv
eg p \land
eg q$, we can rewrite $
eg p \lor
eg q$ as $
eg (p \land q)$. In words, this means "It is false that (the hotel has a weight room and the conference center has an auditorium)". Wait, no, wait. Wait, the given statement is $
eg p \lor
eg q$. Let's re - express:

Wait, the original statement is "The hotel does not have a weight room ($
eg p$) or the conference center does not have an auditorium ($
eg q$)", so it is $
eg p \lor
eg q$. By De Morgan's law, $
eg p \lor
eg q\equiv
eg(p\land q)$. Which means "It is false that (the hotel has a weight room and the conference center has an auditorium)"? Wait, no, let's check the options again.

Wait, option D says "It is false that the hotel has a weight room and the conference center has an auditorium", which is $
eg(p\land q)$, and since $
eg p \lor
eg q\equiv
eg(p\land q)$, so D is equivalent? Wait, no, wait the original statement is $
eg p \lor
eg q$. Let's check each option:

Option A: $p\land q$: not equivalent to $
eg p \lor
eg q$.

Option B: $
eg(p\lor q)$: which is $
eg p \land
eg q$, not equivalent to $
eg p \lor
eg q$.

Option C: $p\land
eg q$: not equivalent.

Option D: $
eg(p\land q)$: and since $
eg p \lor
eg q\equiv
eg(p\land q)$ (by De Morgan's law), so D is the equivalent statement. Wait, but wait the original statement is $
eg p \lor
eg q$, and De Morgan's law says $
eg(p\land q)\equiv
eg p \lor
eg q$. So the equivalent statement is "It is false that (the hotel has a weight room and the conference center has an auditorium)", which is option D. Wait, but in the initial marking, B was marked, but that's a mistake. Let's re - do:

Let $p$: Hotel has a weight room, $q$: Conference center has an auditorium.

Given statement: $
eg p \lor
eg q$.

De Morgan's law: $
eg(p\land q)\equiv
eg p \lor
eg q$.

So the equivalent statement is $
eg(p\land q)$, which is "It is false that (the hotel has a weight room and the conference center has an auditorium)", which is option D.

Wait, maybe I made a mistake earlier. Let's check option D: "It is false that the hotel has a weight room and the conference center has an auditorium" – that is $
eg(p\land q)$, and since $
eg p \lor
eg q\equiv
eg(p\land q)$, so D is correct.

Answer:

D. It is false that the hotel has a weight room and the conference center has an auditorium.