The domain for $x$ and $y$ are all integers.
$∃x∀y (y=x^2+2x+1)$ can be interpreted as: There is an $x$ for all $y$ such that it satisfies $y=x^2+2x+1$. There is a single $x$ value which results in all the domain values of $y$ – all the integers.
This can't be true, as the relationship between $x$ and $y$ is given; a single $x$ value cannot result in every elements in the range of integers. Thus, the truth value of the statement is false.
Yet, a different justification for the false value was stated on the answer sheet. Referring to it, the statement's truth value is false because for every $x$, there is always an integer $y=x^2+2x+2>x^2+2x+1$.
Can anyone explain to me regarding its meaning? Thanks.
Best Answer
The answer sheet is merely pointing out that the negation $$∀x∃y (y\neq x^2+2x+1)$$ of $(*)$ is True, and thus $(*)$ itself must be False.