GATE CS 2015 Set 2 Q.28 (CS Q.18): Sets, Relations, Functions, Partial Orders and Lattices

The cardinality of the power set of $\{0, 1, 2, \dots, 10\}$ is _____.

Your answer

Official answer and solution

Solution

Count the number of elements in the given set and compute $2^n$ .

The given set is:

S = \{0, 1, 2, \dots, 10\}.

The elements are the integers from 0 to 10 inclusive. The cardinality of $S$ is:

|S| = 10 - 0 + 1 = 11.

The power set $\mathcal{P}(S)$ is the set of all subsets of $S$ . The cardinality of the power set of any finite set with $n$ elements is $2^n$ :

|\mathcal{P}(S)| = 2^{11} = 2048.

Final answer

2048

Common trap

Counting 10 elements instead of 11 by omitting the 0 element, which would lead to $2^{10} = 1024$ .

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