Recall powers (or exponents) of numbers:

Similarly, sets have the power operation to create new sets.

If A is a set, then the **power set** of A is another set denoted as

Recall: A is a subset of B if every element in A is also in B. Furthermore, if A is a finite set with n-elements, then we can find the number of subsets in A by using this formula:

To find the power set of A, we write a list of all the subsets of A first – remembering that:

- the empty set is a subset of every set,
- and every set is a subset of itself

Let’s look at some examples:

e.g.1.

Using the formula , we know that there are possible subsets of A, namely:

Hence the power set is the set that contains all the above subsets:

Note: The cardinality (size) of where size of A= 3 elements

e.g.2.

We know there are possible subsets of B, namely:

Hence the power set is:

Note: Again, the cardinality of where set |B| = 2 elements

The cardinality of a power set A, where A is a finite set, is denoted as

i.e.

e.g. Find the cardinality of the following power sets

Set A = in this case. We know we have 1 element in the set (i.e. |A| =1), hence .

The subsets are hence the power set is

2.

Set A = in this case hence, again, we have 1 element in the set (i.e. |A| =1). So subsets and .

The subsets are hence the power set is

3. .

Set A = in this case, hence we have 0 elements in the set (i.e. |A| =0). So .

So the subset is only the empty set, { }, and the power set

4. .

Set A = in this case. We have |A| = 2 = ||, hence subsets and .

Hence the power set is

If a set A is infinite, we cannot find all its subsets, hence cannot list its power set.

e.g. instructs for a power set of an infinite set, the integers: Hence we cannot find the power set. We can, however, list some subsets:

- the empty set,
- the set of integers
- and any subset of integers, for instance

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