Exam review, How did he find the size of AxA?

[mr_coffee]

Hello everyone, another review question I'm lost on,

He has the following question
A binary relation on a set S is defined to be any subset of the Cartesian product S x S. (We will study binary rleations later in the course.) If S is a set of size 3, how many binary relations are there on the set S?

We never went over what a binary relation is, so we must not need it for this question.

But if the size of S = 3, then size of SxS = 9 right?

And if he told us a binary relation on a set S is deifned to be any subset of SxS then he must mean take the power set of (SxS). I know the size of a power set is defined as 2^n, where n is the number of elements. So I would get

2^9 = 512 which is the correct answer.
( I just got this while typing the question out)

But now this brings up another question...

Let A = {2, 3, 5} and B = {3,4,5,6}
If A has 3 elements and B has 4, is there a fast way to figure out the size of AXA or do you have to write out and just count the elements? the answer is 9. So could i have just said well there are 3 elements in A, so 3x3 = 9. Or if i had AXB would that just be 3x4 = 12 elements

 

[quasar987]

Well it's the very essence of multiplication. In the probability book my class uses this is called the "fundamental principle of counting". If there are n ways to pick an element from A and for each of them, there are m ways to pick an element from B, then there are nxm ways to form a pair of 1 element from A and one element from B.

 

[mr_coffee]

ahh thanks for the info!