Number of Pairs of Subsets in X with Unique Elements: Sets and Subsets

[libragirl79]

For a pair (A,B) of subsets of the set X=(1,2,...100), let A*B denote the set of all elements of X which belong to exactly one of A or B. what is number of pairs (A,B) of subsets of X such that A*B=(2,4,6,...100)?



I let A =(1,2,3...50) and B=(51,52,...100) so there are 25 elememnts of even numbers in each set, multiply them, so there are 625 pairs of (A,B).

 

[clamtrox]

I let A =(1,2,3...50) and B=(51,52,...100) so there are 25 elememnts of even numbers in each set, multiply them, so there are 625 pairs of (A,B).

No, that's 1 pair: (A,B). Let's think this more carefully. If A = {1,2,3,...,100} = X and B = {1,3,5,...,99}, then what's A*B?

 

[libragirl79]

I understand that you sitll have A and B as subsets of X, but if A is X and B is only the odd numbers, then how are we supposed to get the pairs of even, wouldn't they come from A only then? Thanks!

 

[clamtrox]

A*B is the set of all elements of X which belong to exactly one of A or B. Let A and B be what I said above. Does 1 belong to A*B? How about 2? How about 3? Try to work it out!

 

[libragirl79]

Well since 1 doesn't, 2 does, 3 doesn't, etc it means that only even ones from A work, so that would be 50 even numbers, but that sounds too simple...

 

[voko]

You need to think how you could build all the possible A, B pairs that satisfy the condition.