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I need help with nested quantifiers that use the implication symbol. This problem is giving me a headache as I cannot seem to find mistakes by reading it back. In particular are the word choices I have to use. Below are some examples and answers.

  1. Let S(x) = “x is a student in our class” and P(x) = “x works at the mall” Assume the u.d. for x is all people.

Translate into symbols, “There is a student in our class who works at the mall.

∃x (S(x) -> P(x))

  1. Let S(x) = “x is a student at Blue College”; F(x) = “x is a faculty member at Blue College”, and E(x,y) “x has eaten y at the Blue College cafeteria”

where the universe of discourse for x is all the people who are associated with Blue College, and the universe for discourse for y is the menu items in the College cafeteria.

Translate into English: ∃x¬∃y(F(x)→(E(x,y))

Some faculty member at Blue College has not eaten some menu items at the Blue College cafeteria.

Translate into symbols: Every Blue college student has eaten at least one menu item at the Blue College cafeteria.

∀x∃y(S(x)→ E(x,y))

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    Do you have to use the implication symbol? Especially in the case of your first sentence it seems that a conjunction is more appropriate. – 2017-02-14
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    I can use other logic symbols like disjunction and conjuction. After reading other replies I think I understand now though, thanks. – 2017-02-14

2 Answers 2

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The sentence

$$\exists x:(S(x)\implies P(x))$$

reads in English as

There exists a person who, if he is in our class, he is working at the mall.

This statement is true for me, because I am not a student in your class, which means that the statement "If 5xum is a student in DOe's class, then he works at the mall" is true. (remember, if $A$ is not true, then $A\implies B$ is always true).

So no, your first solution isn't what you want. I suggest you try again.

Your second attempt is also wrong. Let's say Joe is the only faculty member, and he ate steak, but never ate salad. Then the statement

Some faculty member at Blue College has not eaten some menu items at the Blue College cafeteria.

is clearly true, since Joe never ate salad.

However:

  • The statement $F(Joe)\implies E(Joe, steak)$ is true,
  • Therefore, which means that $\exists y:(F(x)\implies E(x,y))$ is true (because it is true for $y=steak$),
  • Therefore, so the statement $\neg \exists y:(F(x)\implies E(x,y))$ is false (because it is the negation of a true statement).

So, the entire statement must also be false (because it's clearly false if $F(x)$ is not true, and $F(x)$ is true only if $x=Joe$).

Your third answer is correct.

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    Thank you, so to fix that would "Some faculty member at Blue College has not eaten any menu items at the Blue College cafeteria." work? – 2017-02-15
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Implication restricts the universal quantifier.

$~\forall x~[P(x)\to Q(x)]$.   "Every thing that satisfies $P$ will satisfy $Q$."

Conjunction restricts the existential quantifier.

$~\exists x~[P(x)\wedge Q(x)]$   "Some thing that satisfies $P$ also satisfies $Q$."

Don't get them mismatched.