# Thread: Cross Product

1. ## Cross Product

For finite sets M and N and P, if the cross product of M and N is equal to the cross product of M and P, then N = P. (if MxN=MxP then N=P)

my prof told us its FALSE. but why?

2. ## Re: Cross Product

This is also false for numerical product for a similar reason.

3. ## Re: Cross Product

what's the reason?

4. ## Re: Cross Product

First say if you think the following fact is true for numbers:

For all m, n, p, if mn = mp, then n = p.

If you think this is true, try proving it. Pay special attention so that all operations you do in the proof are well-defined.

5. ## Re: Cross Product

Now i get it:

For all m, n, p, if mn = mp, then n = p:

for example

if m = 0, n = 2, p = 100

then: mn = 0, mp = 0 and mn = mp
but n is not equal to p

am i right?

6. ## Re: Cross Product

By the way, when A and B are sets, $A\times B$ is the "Cartesian product", not the "cross product".

7. ## Re: Cross Product

Yes.

Now consider some sets M, N, P with 0, 2, and 100 elements, respectively.

8. ## Re: Cross Product

Now consider some sets M, N, P with 0, 2, and 100 elements, respectively.
M = {0}
N = {2}
P = {100}

MxN = {(0,2)}
MxP = {(0,100)}

how would MxN be equal to MxP?

9. ## Re: Cross Product

Originally Posted by nickgc
For finite sets M and N and P, if the cross product of M and N is equal to the cross product of M and P, then N = P. (if MxN=MxP then N=P) my prof told us its FALSE. but why?
Consider: $\emptyset = \emptyset \times \{ 1\} = \emptyset \times \{ 2\}$

10. ## Re: Cross Product

in ordinary language, if we never pick a first coordinate, it doesn't matter which second coordinate we were going to pick, it never gets chosen.

the same logic goes for the second coordinate: no matter which first coordinate we pick, if we never choose the second coordinate, we never get a pair.

picking an element of the empty set pretty much stops us dead in our tracks, it has no elements to pick.

11. ## Re: Cross Product

Consider: (empty set) = (empty set) x {1} = (empty set) x {2}
OK now i understand it.. thank you very much