Question

If $\mathbf{C}$is a set with$\mathbf{c}$elements, how many elements are in the power set of $\mathbf{C}$? Explain your answer.

Short answer

The number of elements in the power set of $\mathrm{C}$is$2^{\mathrm{c}}$ elements.

Step-by-step solution

Define Power Set

The number of elements in the set $\mathbf{S}$is$\mathbf{n}$, then its power set consists of${\mathbf{2}}^n$elements. The Power set is a set of all subsets of a set including empty set.

For example, the power set of set$\{1,2\}$ is $\{\mathrm{\varphi },\left\{1\right\},\left\{2\right\},\{1,2\}\}$.

Substitute ‘c’ instead of ‘n’

The set $\mathrm{C}$contains$\mathrm{c}$ elements such as$\{{\mathrm{c}}_1,{\mathrm{c}}_2,{\mathrm{c}}_3,\mathrm{...}{\mathrm{c}}_{\mathrm{c}}\}$. By substitutingcinstead ofn,the number of elements in the power set of $\mathrm{C}$is $2^{\mathrm{c}}$elements.

Therefore, the number of elements in the power set of $\mathrm{C}$is$2^{\mathrm{c}}$ elements.