A restaurant asked customers to list their favorite menu items. The table lists the answers of eight customers. Using the table, complete the following:

(a) Choose a set that is equivalent, but not equal, to the set of Dan's favorite menu items.
(Choose one)

(b) Choose a set that is equal to the set of Ahmad's favorite menu items.
(Choose one)

(c) Find two customers whose sets of favorite menu items are neither equivalent nor equal.
(Choose one)
and (Choose one)

\begin{tabular}{|l|l|}
\hline
Customer & \multicolumn{1}{|c|}{ Set of favorite menu items } \\
\hline
Ivan & \{Ribs, Jambalaya, Fries\} \\
\hline
Shen & \{Samosa, Hamburger, Fries, Gumbo\} \\
\hline
Charlie & \{Chowder, Tacos\} \\
\hline
Milan & \{Samosa, Chowder, Hamburger, Tacos\} \\
\hline
Ahmad & \{Fries, Ribs, Jambalaya\} \\
\hline
Eric & \{Hamburger, Meatloaf, Pilaf, Sausage, Lasagna\} \\
\hline
Dan & \{Samosa, Hummus, Brisket, Ravioli, Dumplings\} \\
\hline
Tom & \{Fries, Shrimps, Jambalaya\} \\
\hline
\end{tabular}



Answer :

Sure, let's solve this step by step using the information provided in the table:

(a) Choose a set that is equivalent, but not equal, to the set of Dan's favorite menu items.

To answer this question, we would need a set of items that contains the same elements as Dan's set [tex]\(\{Samosa, Hummus, Brisket, Ravioli, Dumplings\}\)[/tex], but represented differently in some sense. However, from the table and given information, there are no sets that are equivalent (contain the same elements) but not exactly identical with respect to Dan's favorite set. Therefore, the answer here would be an empty set: [tex]\(\emptyset\)[/tex].

(b) Choose a set that is equal to the set of Ahmad's favorite menu Items.

Ahmad’s favorite menu items are: [tex]\(\{Fries, Ribs, Jambalaya\}\)[/tex]. We need to find another customer who has exactly these favorite items. From the table, Ivan’s favorite items are also [tex]\(\{Fries, Ribs, Jambalaya\}\)[/tex].

Hence, the customers with a set equal to Ahmad's are:
- Ivan
- Ahmad

(c) Find two customers whose sets of favorite menu items are neither equivalent nor equal.

To solve this, identify customer pairs whose favorite items are completely different without any subset relationships. From the table, there are multiple pairs that satisfy this condition. One example pair is:
- Ivan (\{Ribs, Jambalaya, Fries\})
- Shen (\{Samosa, Hamburger, Fries, Gumbo\})

Another example pair is:
- Shen (\{Samosa, Hamburger, Fries, Gumbo\})
- Charlie (\{Chowder, Tacos\})

Therefore, one example answer is:
Shen and Charlie

In summary:

(a) No equivalent but not equal sets for Dan: [tex]\(\emptyset\)[/tex]

(b) Sets equal to Ahmad: Ivan, Ahmad (mark both)

(c) Example pairs of customers whose sets are neither equivalent nor equal:
- Shen and Charlie
- There are multiple possible correct pairs in the table.