The problem states to find the negation of the following statement:
There exists a number which is equal to its square.
Original Answer: There does not exist a number which is equal to its square.
My answer: There exists a number which is not equal to its square
Are both the answers above equal? if not then what am I missing? To me both the answers are not equal from a mathematical point of view.
No, they are not the same. "There exists a number equal to its square" means the set of such numbers is not empty. The negation of this is that the set of such numbers is empty, i.e. there does not exist a number equal to its square.
There exists a number equal to its square (1) and there exists a number not equal to its square (2). Since both these statements are true, they can't be negations of each other. So your answer was not correct.
Your answer presupposes the existence of a number. Think of the following example to see the problem.
Statement (1): There exists a unicorn with a yellow fur.
This statement is false. Yet, so is the statement:
Statement (2): There exists a unicorn which has no yellow fur.
Indeed, the correct negation of statement (1) is
Statement (3): There does not exist a unicorn with yellow fur.
The negation of “For some X, P(X) holds” can be stated in several equivalent ways:
Your original answer is the first of these forms and is correct.
Your second answer is incorrect. It is sort of like the third of these forms, but uses “For some X” instead of “For all X”.
Let $P(n):n^2=n$.
Then the question is $Q:\exists n \;\;\text{s.t.}\;\; P(n) \;\;\text{is true}$
$Q$ is true, therefore $\lnot Q:\not\exists n \;\; \text{s.t.}\;\; P(n) \;\;\text{is true}$
is false.
Let $Q':\exists n \;\;\text{s.t.}\;\; P(n) \;\;\text{is false}$
and so $Q'$ is true (and $\lnot Q'$ is false).
Therefore $Q'\ne \lnot Q$.
$\lnot Q$ is the negation of the statement (which is what the problem asks for), and $Q'$ is a negation of the predicate.
This will be helpful to understand negation:
Statement with universal quantifier: $\forall x, p(x)\implies q(x)$.
It's negation is $\exists x, p(x)\land \lnot q(x)$
Statement with existential quantifier: for some $x, p(x)\land q(x)$.
It's negation is $\forall x, \lnot p(x)\lor \lnot q(x)$ which is further equivalent to
$\forall x, p(x)\implies \lnot q(x)$.
Your statement is "for some $x$, $ x$ is a number and $x^2=x$."
It's negation will be "$\forall x$, if $x$ is a number then $x^2\ne x$" which when formulated in language gives "Every number is not the square of itself."
