- There exists an even integer.
Would this be: There is an integer that is not even. $\forall x \neg E(x)$
You managed to get the logic expression correct, but the English should be "All integers are not even", or "No integer is even".
- Every integer is even or odd.
Would this be: There exists an integer that is not even or odd. $∃ x \neg (E(x) \vee O(x))$
Yes, though it is better to say "... neither even nor odd", to clarify that the negation applies to "even or odd", rather than just to "even".
- All prime integers are non-negative.
Would this be: There exists a prime that is not non-negative. $∃ x \neg P(x) \Rightarrow N(x)$
The English is parseable, but the FOL is not. You want :
$$\underbrace{\exists x }_\text{There is an integer}(\underbrace{P(x)}_\text{that is prime}\;\underbrace{\wedge}_\text{and}\; \underbrace{\neg N(x)}_\text{not non-negative})$$
Remember that the negation of an implication is a conjunction with a negation (of the consequent). $$\;\neg (A\implies B) \;\equiv\; A\wedge \neg B\;$$
- The only even prime is 2.
Would this be: The only even prime is not 2. $∀x \neg(P(x) \wedge E(x)) \Rightarrow 2$
No. The statement is actually short for "2 is an even prime and there exists no other even prime." So the negation would be "Either 2 is not an even prime, or there exists an even prime which is not 2".
$$\neg (P(2)\wedge E(2)) \vee \exists x(P(x)\wedge E(x)\wedge (x\neq 2))$$
Also, you can't simply state something implies $2$. That's nonsensical; $2$ is not a boolean value. You need to equate something with it (or deny the equality as the case may be).
- Not all integers are odd.
Would this be: All integers are odd. $∀xO(x)$
Yes, it would.
The first part is the necessary condition. The second part is the "but not sufficient" condition
The claim is that $(1)$ if the berries are ripe or bears have been seen, then it's not safe and $(2)$ even if the berries are not ripe and bears have not been seen, it still might not be safe (for example, there might be a hurricane approaching). In other words, condition $(1)$ is necessary for safety, but not sufficient to guarantee it.
The statement "$B$ is necessary for $A$" can be expressed as $A\rightarrow B$. The statement "$B$ is sufficient for $A$" can be expressed as $B \rightarrow A$. The statement "$B$ is necessary but not sufficient for $A$" can be expressed as $(A \rightarrow B) \land \lnot(B \rightarrow A)$.
Thus, in your proposed version, you've only indicated the necessary condition. I agree, your version doesn't claim it's sufficient. But the statement in the book says more: It explicitly says that the necessary condition is not sufficient.
Best Answer
You're right - $\wedge$ is the correct interpretation. English statements such as "even though", "however", "but", and "yet" all have usages that contrast facts, but they are all translated to "and" in propositional logic.