Expressing a set in tabular method Set $A=\\{\,x \in\mathbb R : x^2-(a+b)x +ab=0\,\\}$. We have to express it using tabular method. At first, we factorize it, getting $(x-a)(x-b)=0$ or $x =a$ and $x=b$. But how do we know that $a$ and $b$ belongs to $\mathbb R$? Is there any proof of this fact, or are we just assuming it?

Unless it is _given_ (e.g. in the problem statement) that $a,b\in\mathbb R$, you can't assume it. So if $a=1+i\
otin \mathbb R$ and $b=7\in \mathbb R$, your tabular method should yield $\\{7\\}$, not $\\{1+i,7\\}$. If nothing about $a,b$ is specified, you'd have to answer with listing various cases (according to if/which/how many of $a,b$ are real).