P is the set of "all exams at university". E(x) denotes the predicate "x is easy". Translate the following sentence to formal logic. "Only one exam is easy." Solution:
I understand the first part, that there exists a test "x" for which the predicate "is easy" holds. But I don't understand the second part. I read it as such that for all y,z element of the set of exams, if y is easy and z is easy and there exist an exam x that is easy it follows that y = z.
On
The appropriate translation "Only one exam in the university is easy" is
$$\exists x \in P\Big(E(x) \land \forall y\in P(E(y) \to (x=y))\Big).$$
The translation you posted says there does exist an easy exam, and for all exams, if they are both easy, they are equal.
On
The second part says that if you pick any two easy exams, then they will turn out to be one and the same exam. This will rule out that there are two (or more) different easy exams. Indeed, you could equivalently write that second part as:
$$\neg \exists y, z \in P: (E(y) \land E(z) \land y \neq z)$$
Finally, here's an efficient way to say that there is exactly one easy exam:
$$\exists x \in P: \forall y \in P: (E(y) \leftrightarrow x = y)$$
The first part says: "some exam is easy".
The second part says: "if there are two easy exams, they are equal".
See Uniqueness quantification.