# Multiple quantifiers questions - discrete mathematics?

So I have some questions below that I don't understand because I'm struggling with solving questions that involve multiple quantifiers. I was wondering if someone could walk me through how to do these? I need help with especially 2 because I'm really lost as to how to approach that question.

For 1, I think I can solve it if there wasn't the added statement added 'K(x, y) denote "x knows y"'. I'm not sure how that changes the answer.

1. Let H(x) denote "x is a hockey player", C(x) denote "x is a hockey coach", O(x) denote "x is a person in Ottawa", and K(x, y) denote "x knows y". State the universe of discourse and translate the following proposition into a predicate logic formula: "Every hockey coach in Ottawa knows at least one hockey player in Ottawa."

2. Let the universe of discourse be { a, b, c }. Write out the following propositions explicitly so that they do not contain any universal or existential quantifiers:

a. ∀x Ǝy F(x, y)

b. Ǝx ∀y F(x, y)

c. ¬(∀x Ǝy F(x, y))

I'm also confused by the universe of discourse being { a, b, c } in 2.

### 1 Answer

- az_lenderLv 71 year agoFavorite Answer
#1. I can't type the upside-down A or the backwards E, so I'll just type A and E. The desired quantified statement is:

(Ax)((C(x) ^ O(x)) => ((Ey)(O(y) ^ H(y) ^ K(x,y)))).

#2. (a)

(F(a,a) v F(a,b) v F(a,c)) ^ (F(b,a)v F(b,b)v F(b,c)) ^

(F(c,a)v F(c,b)v F(c,c)).

(b) (F(a,a)^F(a,b)^F(a,c)) v (F(b,a)^F(b,b)^F(b,c)) v (F(c,a)^F(c,b)^F(c,c)).

(c) Copy the answer to (a) above and enclose it in parentheses, and then put a negation sign to the left of it.

Someone put a thumbs down, but I (the responder) have no idea what their beef was...