With the set $X=\{1,2,...,n\}$, How many subsets of $X$ contain $1$? How many subsets of $X$ don't contain $1$?

My work: 
I know a set with $n$ elements has $2^n$ subsets. The way I tackled this problem was to make a finite set like {1,2,3} and tried to make some generalization from it. I made a decision tree diagram to figure out the different subsets. 

If $1$ was contained in the subset, then the different subsets with $1$ would be:
$$\{1,2,3\},\{1,2\},\{1,3\},\{1\} $$ 
which made me think there are four subsets. $2^n = 4$ which means $n=2$ but there are 3 elements in here so I generalized it as:
$$2^{n-1}$$
where $n \ge 1$ such that $n \in N $ is how many subsets of X that contain $1$.

And for the subsets of $X$ that don't contain $1$, the answer should come out the same as the previous question because, again, if I were to make a finite set like $\{1,2,3\}$, then the subsets without $1$ would be:
$$\{2,3\}, \{2\}, \{3\},  \{\emptyset\}  $$

which is also, $$2^{n-1}$$
where $n \ge 1$ such that $n \in N $ is how many subsets of X that don't contain $1$.

I think it makes sense in my head but I am still unsure. Am I understanding this problem correctly or am I completely missing it? 


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