This exercise refers to Example. Determine whether each of the following statements is tru...

This exercise refers to Example. Determine whether each of the following statements is true or false.

a. V students S, 3 a dessert D such that S chose D.

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b. V students S, 3 a salad T such that S chose T.

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c. 3 a dessert D such that V students S, S chose D.

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d. 3 a beverage B such that V students D, D chose B.

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e. 3 an item I such that V students S, S did not choose I.

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f. 3 a station Z such that V students S, 3 an item I such that S chose I from Z.

Example

Interpreting Multiply-Quantified* Statements

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 beverage station offers milk, soda, or coffee. 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: spaghetti, fish, pie, soda

These choices are illustrated in Figure.

Figure

Write each of following statements informally and find its truth value.

a. ∃ an item I such that ∀ students S, S chose I.

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b. ∃ a student S such that ∀ items I, S chose I.

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c. ∃ a student S such that ∀ stations Z, ∃ an item I in Z such that S chose I.

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d. ∀ students S and ∀ stations Z, ∃ an item I in Z such that S chose I.

Solution

a. There is an item that was chosen by every student. This is true; every student chose pie.

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b. There is a student who chose every available item. This is false; no student chose all nine items.

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c. There is a student who chose at least one item from every station. This is true; both Uta and Tim chose at least one item from every station.

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d. Every student chose at least one item from every station. This is false; Yuen did not choose a salad.