Quantified statement logic question, descrete math, wee

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The discussion centers on a quantified statement logic problem involving a college cafeteria line with four stations: salads, main courses, desserts, and beverages. Participants analyzed the statement "There exists a station Z such that for all students S, there exists an item I such that S chose I from Z." The conclusion drawn is that the statement is true, as all students selected items from the desserts, main course, and beverage stations, satisfying the logical conditions. The clarification provided emphasizes the importance of the order of quantifiers in determining the truth of the statement.

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mr_coffee
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Hello everyone, i want to see if i did this correctly.

THe question refers to this:

A college cafeteria line has four stations: salads, main courses, desserts, and beverages. The salad station offers a choice of green salad or fruit salad; the main course station offers spaghetti or fish; the dessert station offers pie or cake; and the bevrage station offers milk , soda, or cofee. Three students, Uta, tim and Yuen, go through the line and make the following choices:

Uta: green salad, spaghetti, pie, milk
Tim: fruit salad, fish, pie, cake, milk, coffee
Yuen: spagehtii, fish, pie, soda

Determine wheher each of the follow statement sis true or false.

f. [tex]\exists[/tex] a station Z such that [tex]\forall[/tex] students S, [tex]\exists[/tex] an item I such that S chose I from Z.

I believe this statement says: IN a particular station Z, All students S chose a particular item I from a station.

I said True, every student S, chose pie from a particular station Z (desserts Station).

Thanks!
 
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It's true, but not just because of that. The statement means:
There is a station such that every student chose some item from the station.
This is satisfied by the desserts station, but also by the main course station and the beverage station. If you swapped the order of "for all students S" and "Exists an item I" then it would only be satisfied by the desserts station, but as it is the items the students choose from the station do not have to be the same, so long as they each choose some item.
 
Ahh i c now! thanks for the help! again!
:)
 

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