# Logical quantifiers help please

Let $T(x; y)$ denote the phrase "$x$ likes cuisine $y$", where the domain of $x$ is the set of students and the domain of $y$ consists of all cuisines. Express each English sentence below in terms of $T(x; y)$ with quantiers.

Alice and Bob have the same taste. That is, any cuisine Alice likes, Bob likes and vice versa.

Except for Chinese cuisine, Carla and Dave have opposite tastes. That is, except for Chinese food, any cuisine Carla likes, Dave dislikes and vice versa.

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Please Help if you have any idea – Alpha Oct 23 '12 at 2:16
specially for part b and c – Alpha Oct 23 '12 at 2:17

In many ways, the logical symbols just provide a compact representation of a clear statement of the logical proposition/sentence/etc. you're trying to express.

So for:

Alice and Bob have the same taste. That is, any cuisine Alice likes, Bob likes and vice versa.

We can say something like "for every cuisine, Alice likes it if and only if Bob likes it", and we start to get something that's very easy to translate (I'm going to label the set of set of cuisines with $C$ for convenience): $$\forall y \in C\;(T(Alice;y)\Leftrightarrow T(Bob;y))$$ Similarly, by clearly writing out the cases for the second sentence:

Except for Chinese cuisine, Carla and Dave have opposite tastes. That is, except for Chinese food, any cuisine Carla likes, Dave dislikes and vice versa.

We can get an English phrase that's easy to translate; "for every cuisine, either it is Chinese, or Carla likes it and Dave doesn't, or Carla doesn't like it and Dave does". $$\forall y \in C\; ((y=\text{Chinese}) \vee (T(Carla;y)\wedge\neg T(Dave;y) \vee (\neg T(Carla;y)\wedge T(Dave;y)))$$

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Thank Man ,,, Awesome answer, Thanks – Alpha Oct 23 '12 at 3:03

For your first question, how would you assert that $T(\text{Alice}, y)$ and $T(\text{Bob}, y)$ have the same truth value for all cuisines $y$? Start by providing an expression that will be true for some particular but unspecified $y$ and quantify that.

For the second question you have to assert $T(\text{Carla}, y)$ and $T(\text{Dave}, y)$ have opposite truth values for all cuisines $y$ except when $y$ isn't Chinese (presumably, we don't care about what the truth values are in that case). Do you see how to go from there?

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Thanks Man ,,, , – Alpha Oct 23 '12 at 3:04