Heres the problem
Heres my attempt. I'm not sure if I have done it correctly, if not, am I on the right track?
Edit: My answer is meant to be true, not false
Although that is no absolute set of symbols in mathematics some notations are widely accepted. The symbol $\displaystyle \mathbb{R}$ is used to denote the real numbers. $\displaystyle \mathbb{R}^+$ denotes the positive reals and $\displaystyle \mathbb{R}^-$ denotes the negative reals.
Thus $\displaystyle \mathbb{R}= \mathbb{R}^+ \cup \mathbb{R}^- \cup {0}$ so the nonnegative reals are $\displaystyle \mathbb{R}\backslash \mathbb{R}^ - = \mathbb{R}^ + \cup \{ 0\} $.
As to the “Was I totally wrong?”, I really have no idea what you did. I cannot follow it. The symbolic statement reads “There is some nonnegative real number, x, having the property that for every real number y then $\displaystyle x^2<y+1$.
You see that zero has that property; so does $\displaystyle \frac {1} {2}$.