A basic question related with compactness I have to check for compactness of given subets of $\mathbb{R}^2$.
$A =\{(x, y) :xy = 1\}$
$B =\{(x, y) :x^2y^2 = 1\}$
$C =\{(x, y) :e^x = \cos y\}$
$D =\{(x, y) :\mid x\mid +\mid y \mid \leq 10^{100}\}$
The purpose of asking above question is not just to get answers. I need concepts to deal with these kind of problems. Let me explain where I face difficulties. 
Take set $A$; Intuitively this is clear to me that  $A$ is not a compact subset of $\mathbb{R}^2$  as it is closed  but unbounded. My problem is I am having trouble with checking boundedness or unboundedness of given subsets. Here I know that set $A$ consists of points which lies on rectangular hyperbola. So I have no difficulty in judging that set $A$ is unbounded. But I am not sure about others. Since I am not able to figure out set them.
Edit: I don't want graphical approach to solve these problems. Because quite often i face problems where I find myself unable to visualize graph of given functions. I think there must be available some mathematical tool to deal with this.
I need help to understand this. I would be very much thankful to all of you.
 A: For this you need the Heine-Borel Theorem which says that a subset of $\mathbb{R}^n$ is compact iff it is closed and bounded.
For $B$, note that the positive branch of $xy = 1$ is a subset of $B$. Since it is not bounded, $B$ fails to be compact.
For $D$ the constant $10^{100}$ is a red herring; the result is the same for any positive constant.  Draw what $|x| + |y| \le 1$ and you will get the idea quickly.
For $C$, look at the graph of $x=\log(\cos(y))$
A: Draw pictures, or at least visualize. In set $B$, $x$ can be arbitrarily big. If you pick any $x\ne 0$, there is a $y$ such that $x^2y^2=1$. So there is no disk with centre the origin that contains all of $B$.   
Also in set $C$, $x$ can be arbitrarily large negative. Just choose $y$ close to $\pi/2$, but a tiny bit below.  Or else note that $y$ can be anything that makes $\cos y$ positive, and there are arbitrarily large $y$ with this property, such as $y=2n\pi$ where $n$ is any positive integer.  
The set $D$ is clearly bounded, it is inside the disk with centre the origin and radius $10^{100}$.  So you need to check whether or not $D$ is closed.
