What is the Power Set of the Empty Set? | Set Theory

What is the power set of the empty set? We will answer this question in today’s math lesson! We will write the empty set like so: { }. Recall that the power set of a set A is the set containing all subsets of A. So, for example, P({ 1 }) = { { }, { 1 } }. Also, recall that if the cardinality of a set A is n, that is |A| = n, then the cardinality of the power set of A is 2^n, that is |P(A)| = 2^n.

So, since the empty set has 0 elements, we know |P({ })| = 2^0 = 1. There is one element in the power set of the empty set, and that one element is the empty set. So, P({ }) = { { } }. The power set of the empty set is the set containing the empty set because the empty set is the only subset of the empty set.

We know no other set can be a subset of the empty set because there are no elements in the empty set, so there is no set whose elements are entirely contained in the empty set except for the empty set itself. Again, the empty set is the only subset of the empty set, so the power set of the empty set is the set containing the empty set. That is, P({ }) = { { } }.

I hope you find this video helpful, and be sure to ask any questions down in the comments!

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