Why does a set of m elements have 2$^m$ subsets?

Question

Note: This example is from Discrete Mathematics and Its Applications [7th ed, prob 2, pg 576], shout out to @crash.

I understand why $A \times A$ has $n^2$ elements(because every member of set $A$ can form a combination with itself and every other element of $A$. Can someone explain the mathematical intuition of how a set with $m$ elements has $2^m$ subsets?

Answer 1

A element in the set can be either included or not in the subset. So you have 2 possibilities for each element, which equals $2^m$ possibilities, or subsets

Answer 2

Hint: how many subsets of $0$ elements does $A$ have? How many subsets of $1$ element? How many subsets of $m$ elements? $$2^m = (1+1)^m = \sum_{k=0}^m {m \choose k}.$$