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Let S be a set of people, C be the set of all countries, and let T be a predicate defined over S×C such that T(x,y) is True if and only if x∈S has traveled to country y∈C. I need to express each of the following statements by a simple English sentence that sound natural.

a) $(\exists x \in S, T(x, France)) \wedge (\forall y \in S, T(y, Japan))$

b) $ \forall x \in S, \exists y \in C, T(x, y)$

c) $\forall x,z \in S, \exists y \in C, T(x, y) \leftrightarrow T(z,y)$

My answers are the following:

a) Someone has traveled to France and everyone has traveled to Japan.

b) Everyone has traveled to at least one country.

c) Everyone has traveled to the same countries.

Have I translated them correctly? should I be fixing anything to make them sound more natural? Also if I were to define T(x, y) formally, would it be $$T(x,y) : \text{"True iff x has traveled to country y" where } (x, y) \in S \times C$$ or should the domain for the arguments be $x \in S, y \in C$

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Your translations for a) and b) are correct, but not for c)

The logical representation of "Everyone has traveled to the same countries" would be:

$$\forall x,z \in S, \forall y \in C, T(x,y) \leftrightarrow T(z,y)$$

But your given statement has an existential quantifier for the countries, rather than a universal:

$$\forall x,z \in S, \exists y \in C, T(x,y) \leftrightarrow T(z,y)$$

This latter logic statements translates back to English as something like:

"For every pair of people, there is some country that either both of those people have traveled to, or that they both did not travel to"

Let me give an example to illustrate the difference:

Suppose we have three people: Alice, Bob, and Carla

Also supose we have three countries: France, Japan, and Germany

Suppose Alice traveled to both France, Japan, and Germany

Suppose Bob only traveled to Japan

Suppose Carla traveled to Germany

With that, the claim

$$\forall x,z \in S, \exists y \in C, T(x,y) \leftrightarrow T(z,y)$$

is true, because:

for the pair Alice+Bob, we can point to Japan as a country that they both traveled to

for the pair Alice+Carla, we can point to Germany as a country that they both traveled to

for the pair Bob+Carla, we can point to France as a country that they both did not travel to

But clearly, the statement:

$$\forall x,z \in S, \forall y \in C, T(x,y) \leftrightarrow T(z,y)$$

is not true ... that would only be true if the set of countries for all people would be exactly the same (e.g. Alice, Bob, and Carla all traveled to France and Japan, but not to Germany)

Your second question:

Also if I were to define T(x, y) formally, would it be $$T(x,y) : \text{"True iff x has traveled to country y" where } (x, y) \in S \times C$$ or should the domain for the arguments be $x \in S, y \in C$

Saying $(x, y) \in S \times C$ or that $x \in S, y \in C$ is the same thing, both are fine. However, I am more concerned about the $\text{"True iff x has traveled to country y"}$ part: you should leave that simply as:

$$T(x,y) : \text{"x has traveled to country y"}$$

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