Subsets of the power set

Question

Consider the set $$ \mathcal{A}\equiv\{0,1,...,J\} $$ with $J\geq 1$.

I'm struggling to compute:

1) the number of subsets of $\mathcal{A}$ containing $0$

2) the number of subsets of $\mathcal{A}$ containing $1$

Could you help?

Answer 1

Any subset of $A$ containing $0$ takes the form $\{0\}\cup B$, where $B\subseteq \{1, \dots, J\}$. The other direction is also true: $\{0\}\cup B$ for $B\subseteq \{1, \dots, J\}$ is always a subset of $A$ containing $0$. Therefore, $T\subseteq A$ contains $0$ if and only if it can be written as $T=\{0\}\cup B$ for $B\subseteq \{1, \dots J\}$. Hence, the question is equivalent to asking how many subsets $B$ of $\{1, \dots, J\}$ exist. $\{1, \dots, J\}$ has $J$ elements, so the answer is $2^J$. The solution for $1$ is similar.