Prove that product quantities are equal?
Hello,
to show the equality of the following sets:
First, I interpreted the quantities on either side of the equation as M and N. Therefore, for the two quantities to be equal, the following must hold:
I then assumed that the tuple T was an element of the set M (in this case, the set on the left in the equation). This yields:
For the elements x and y of the tuple T we get:
At this point it becomes clear to me that x and y can both exist in only one of the two combined sets in order for the equation to be satisfiable under the previously made assumption. If x is an element of A , then it must not also be an element of C. In addition, in this case y must be an element of D , but not of B. If x is an element of C and not of A , then y must be an element of B , but not of C. These two possible cases are logically expressed by the right-hand side of the equation.
I hope you understand my train of thought. However, I'm having a little trouble mathematically proving what I described above.
I look forward to your advice and please correct me if I misunderstood something!
(2 votes)
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so to the left of the equal sign is:
Now you could make a truth table… or apply some other calculation rules that I can't think of right now…
the same for the right side…
or?
Thank you for your detailed formalization, very helpful!
dange für den Stern… *hüpf* 🫡
If x is an element of A, y must not be an element of B (otherwise (x,y) would be an element of A x B).
This means that y is automatically an element of D (it must lie in the union B and D).
This means that x is not an element of C (otherwise (x,y) would be an element of C x D).
Couldn't x be from C and y from either B or D? So, a kind of XOR?
What I wrote above applies if x is from A. Of course, in general, x is only from the union of A and C, which means that one must also consider the case where x is not from A. The argument can be made analogously:
Then x is in C (it must lie in the union A and C).
Then y must not be an element of D (otherwise (x,y) would be an element of C x D).
This means that y must be an element of B (it must lie in the union B and D).
Thus (x, y) is an element of (C \ A) x (B \ D).