While writing following statement symbolically I ran into confusion. U is the set of all integers. 1) All the primes are non-negative let P(x) is used to express primes and N(x) for non negatives then i came up with two solutions
1) ∃x (P(X) ∧ N(X))
2) ∃x (P(X) → N(X))
My Teacher says the second one is better but she cant explain the reason.Can someone explain
Your teacher cannot explain why the second is better because neither given expression is right; nor is either really less wrong that the other.
This translates directly to a universally quantified statement: "If any integer is a prime, then it is non-negative"
$$\forall x\; (P(x)\to N(x))$$
That is close to your second attempt, except for the quantification. If we are required to use existential quantifier instead of the universal, we apply dual negation.
$$\neg \exists x\;\neg(P(x)\to N(x))$$
Now, the negation of an implication is $\neg(A\to C) \equiv (A\wedge \neg C)$, (via implication equivalence and de Morgan's rule), so: $$\neg \exists x \;(P(x)\wedge \neg N(x))$$
Which translates to "No integer is both prime and negative."
Which is close to your first attempt, except for the negation.
Thus both attempts are wrong. The first is missing two negations, and the second has the wrong quantifier.