Question

If a set has $n$ elements, then it has subsets.

• A. $n$
• B. $2^n$
• C. $2^n-1$
• D. $2^{n-1}$

Answer 1

The correct option is B $2^n$

If a set $A$ has one element, say $A=\left\{a\right\}$, then the only subsets of $A$ are $\left\{\right\}$ and $\left\{a\right\}$. Thus, the set $A$ has $2=2^1$ subsets.
If a set $A$ has two elements, say $A=\{a,b\}$, then the subsets of $A$ are $\left\{\right\}$, $\left\{a\right\}$, $\left\{b\right\}$ and $\{a,b\}$. Thus, the set $A$ has $4=2^2$ subsets.
If a set $A$ has three elements, say $A=\{a,b,c\}$, then the subsets of $A$ are $\left\{\right\}$, $\left\{a\right\}$, $\left\{b\right\}$, $\left\{c\right\}$, $\{a,b\}$,$\{b,c\}$, $\{c,a\}$ and $\{a,b,c\}$. Thus, the set $A$ has $8=2^3$ subsets.
Similarly, if a set $A$ has $n$ elements, then $A$ will have $2^n$ subsets.