List all the subsets of S. You may use "C" for chocolate, "V" for vanilla, and "M" for mint. S = {Chocolate, Vanilla, Mint}
I do not understand how to do this.
{}
{C}
{V}
{M}
{CV}
{CM}
{VM}
{CVM}
To list all the subsets of S = {Chocolate, Vanilla, Mint}, we need to consider all possible combinations of the elements in S, including the empty set.
One way to approach this is by using a binary representation of the subsets. We can assign a binary digit (0 or 1) to each element in S, where 1 represents that the element is included in the subset, and 0 represents that the element is not included.
In this case, we have three elements in S, so we can use a three-digit binary representation. Starting from 000 (which represents the empty set), we can incrementally change the digits to 1 to generate all possible subsets.
Here's a breakdown of the process:
1. Start with 000, which represents the empty set.
2. Increment the rightmost digit:
- 001 represents the subset {Mint}
- 010 represents the subset {Vanilla}
- 100 represents the subset {Chocolate}
3. Increment the middle digit:
- 011 represents the subset {Mint, Vanilla}
- 101 represents the subset {Chocolate, Mint}
- 110 represents the subset {Chocolate, Vanilla}
4. Increment the leftmost digit:
- 111 represents the subset {Chocolate, Vanilla, Mint}
So, the subsets of S = {Chocolate, Vanilla, Mint} are:
- {}
- {Mint}
- {Vanilla}
- {Chocolate}
- {Mint, Vanilla}
- {Chocolate, Mint}
- {Chocolate, Vanilla}
- {Chocolate, Vanilla, Mint}
Note that the order of the elements within each subset doesn't matter, and each element can either be included or excluded.