Set Theory: How Many Elements are in the Cartesian Product of Two Sets?

[Ed Quanta]

If E has m elements and F has n elements, how many elements does E x F have?

My thinking is that E x F would either have m or n elements. If m= n, then E x F would have m elements (or n elements). If m>n, then E x F would have n elements since E x F ={(x,y): x is an element of E and y is an element of F}. This of E has more elements than F, then there could only be n (x,y) pairs since there would only be n y's.Am I correct?

Also, if E x F is an empty set, then it would follow that E or F would have to be an empty set,right?

 

[fourier jr]

think of it as finite sets of points in the plane. the set MxN would be a "rectangle" with mn points. it's true even if M & N are infinite but that's a bit more complicated.

 

[matt grime]

Did it not occur to try a couple of examples, say with small sets with a couple of elements in each?

 

[Ed Quanta]

Thanks Fourier

 

[Ed Quanta]

Can someone provide me with an example where the cartesian product
E x F is a subset of G x H but it does not follow that E is a subset of G and F is a subset of H?
I have been trying to come up with an example that satisifies this condition for the last day or so.
I can see why it would be true that E is a subset of G and F is a subset of H would imply E x F is a subset of G x H. But I am not sure about the reverse direction.

 

[Muzza]

Try letting E or F be the empty set.

 

[mathwonk]

Actually the answer to your question is tautological, i.e. mn is by definition the number of elements of a set MxN where m = card(M) and n = card(N).


In other words the set MxN is considered by many as more basic than the notion of multiplication.


This not so stupid as it appears since the product set makes sense for infinite sets whereas the product of cardinal numbers does not, a priori.

But one even more basic thing has puzzled me all these years: what is a set?

 

[fourier jr]

But one even more basic thing has puzzled me all these years: what is a set?

suppes says it's something that either has elements in it, or is the empty set

 

[HallsofIvy]

suppes says it's something that either has elements in it, or is the empty set

So a set is something that has elements- or doesn't?


I'm moving this to the set theory forum.

 

[fourier jr]

So a set is something that has elements- or doesn't?
I'm moving this to the set theory forum.

lol he says "y is a set <--> there exists x such that x is an element of y or y is the empty set" (p.19)