Introduction to Set Theory

Q1: What is a set?
A1: A set is a collection of elements.
For example, the set of outcomes for a coin toss may be represented as:
$A=\{H, T\}$
where A is the set of outcomes including heads and tails.

Q2: What is the size of a set?
A2: The size of a set is the number of elements it contains.
For the coin toss example, it was clear that $A$ represented a finite set of outcomes.
Moreover, |A|=2. That is the size of A is 2.

However, you can consider other sets, like the set of integers $\Bbb Z={..., -2, -1, 0, 1, 2, ...}$.
Not only is the set of integers an infinite set, but it is countably infinite. Like the finite set of coin tosses, you could go about counting these elements in a systematic way, even if there are an infinite amount of elements.

On the other hand, you could look at the unit set interval $S=[0,1]$, which includes all real numbers greater than or equal to 0 and less than or equal to 1. Like the set of integers, the unit set interval has an infinite number of elements. But there is no conceivable by way by which you could attempt to count the elements in the unit set interval. Hence, it is an uncountable infinite set. Yet we can still associate a size to it: $|S|=1$. In this context, the size of the unit set interval refers to its length, and not the number of elements it contains.