Binary relations on a set

Question

I have a homework problem that asks this...

a) List all the different binary relations on the set $\{0,1\}$

I assume that since the relation is not given then the answer must be the graph, or Cartesian product of the set. This only provides 4 sets. I should be getting this for a solution but I cannot see how to get there. Any explanation would be appreciated.

1. $\emptyset$
2. $\{(0, 0)\}$
3. $\{(0, 1)\}$
4. $\{(1, 0)\}$
5. $\{(1, 1)\}$
6. $\{(0, 0), (0, 1)\}$
7. $\{(0, 0), (1, 0)\}$
8. $\{(0, 0), (1, 1)\}$
9. $\{(0, 1), (1, 0)\}$
10. $\{(0, 1), (1, 1)\}$
11. $\{(1, 0), (1, 1)\}$
12. $\{(0, 0), (0, 1), (1, 0)\}$
13. $\{(0, 0), (0, 1), (1, 1)\}$
14. $\{(0, 0), (1, 0), (1, 1)\}$
15. $\{(0, 1), (1, 0), (1, 1)\}$
16. $\{(0, 0), (0, 1), (1, 0), (1, 1)\}$

This is from an MIT OCW course which I am taking as an independent study as a high school junior.

Answer 1

You have four choices. $0$ can be related to $0$ or not. $0$ can be related to $1$ or not. $1$ can ... etc. until $1$ can be related to $1$ or not. The total number of options is therefore $$2*2*2*2=2^4=16$$