Definition for the set of Real Numbers Could the set of Real Numbers be defined as \begin{array}{l}
\mathbb{R} \equiv \mathbb{Q} \cup \{ x\neq \frac{a}{b} :a\wedge b\in \mathbb{Q} \}
\end{array} ? Why or why not?
 A: The set of cats is not a rational number. So $x$ could be the set of cats. That's probably not a real number.
In slightly more serious terms, where do these $x$ belong? You need a set to draw these from.
Read about dedekind cuts for one approach that derives the reals from subsets of rationals. Cauchy's completeness  argument defines them uses sequences of rationals. These are the classical approaches. The point is we build them from things we know, rather than just say what they are not. 
A: Context is everything.
What is the set $\{x\mid x\neq 0\}$? If you are working in $\Bbb Q$ the set is different from the set you obtain if you work in $\Bbb R$ or in $\Bbb Z$ or in $\Bbb C$.
If you are working inside "the mathematical universe", and you can "access" every mathematical object, then this is not a set anymore, since the mathematical universe is not a set; and moreover the meaning of the symbol $0$ is now unclear.
The same goes in your question. What is the context for $x$ in $x\neq\frac ab$? Is $x$ already a real number? Is $x$ just an arbitrary mathematical object? It's not clear at all. Especially since if you want to define the real numbers then they cannot be the context in which you're working in. The whole idea of defining something is to "refine the given context".
Finally, there is another very mild issue as to whether or not $\Bbb Q\subseteq\Bbb R$, or is there just "a canonical copy of $\Bbb Q$ inside $\Bbb R$". And this will affect the answer as to $\Bbb R$ is equal to that union or not. But let's ignore that for now. 
The real problem is that unless you already assume that $x$ is a real number, there's no way you can claim that this is a good definition for $\Bbb R$; but if $x$ is already a real number, you already had to define $\Bbb R$.
