Here is the sketch representing the set $E$ for $a = c = 1$ and $b = d = 4$. You can see the combination of the circular annulus and the regions between the two hyperbolas.

Let me know if you'd like further details or have any questions!

Follow-up Questions:

1. How do the hyperbolas change if the values of $a$ and $b$ are different?
2. What happens to the shape of the set if we increase the radius of the inner circle (i.e., change $c$)?
3. How do we interpret the union operation in the context of geometric shapes like circles and hyperbolas?
4. What would happen if we included the condition $|xy| = 0$ in the set definition?
5. How would this set look if we used inequalities like $x^2 + y^2 \leq 4$ and $|xy| \geq 1$?

Tip:

When sketching unions of sets like this, it's useful to break down each condition into separate regions and draw them step by step, starting with the basic geometric shapes like circles and hyperbolas, and then refining the sketch based on the conditions.