# Finding all the subsets of a set

I need an algorithm to find all of the subsets of a set where the number of elements in a set is `n`.

```S={1,2,3,4...n}
```

Edit: I am having trouble understanding the answers provided so far. I would like to have step-by-step explanation of how the answers work to find the subsets.

For example,

```S={1,2,3,4,5}
```

How do you know `{1}` and `{1,2}` are subsets?

Could someone help me with a simple function in c++ to find subsets of {1,2,3,4,5}

## Answer

It’s very simple to do this recursively. The basic idea is that for each element, the set of subsets can be divided equally into those that contain that element and those that don’t, and those two sets are otherwise equal.

• For n=1, the set of subsets is {{}, {1}}
• For n>1, find the set of subsets of 1,…,n-1 and make two copies of it. For one of them, add n to each subset. Then take the union of the two copies.

Edit To make it crystal clear:

• The set of subsets of {1} is {{}, {1}}
• For {1, 2}, take {{}, {1}}, add 2 to each subset to get {{2}, {1, 2}} and take the union with {{}, {1}} to get {{}, {1}, {2}, {1, 2}}
• Repeat till you reach n