In the set-builder notation (rule method),the null set is represented by:

Which of the following is a set? Justify your answer.
The collection of all the months of a year beginning with the letter $J$.

Make correct statements by filling in the symbols $\subset$ or $\not\subset$ in the blank spaces:
${ x:x \text{ is an equilateral triangle in a plane} } \dots { x:x \text{ is a triangle in the same plane} }$

Which of the following is an empty set?

What is the remainder when the polynomial $P(x) = 1 + x + x^3 + x^9 + x^{27} + x^{81} + x^{243}$ is divided by $x - 1$?

Determine the subset for the set $\{n(n+1)(2n+1) : n \in \mathbb{Z}\}$.