Proof of Set Identity
Show that if (A∩C)=(B∩C) then A=B, where A, B, C are sets.
*by using a 2- way containment method without using set identities.
*2-way method meaning showing that two sets are equal by showing they contain each other
for example, D=E (where D=(A∩C) and E=(B∩C)) iff D⊆E ^ E⊆D.
so i would split these two up (proof by cases) and prove:
firstly i) D⊆E by proving (x∈D→x∈E)
secondly ii) E⊆D by proving (x∈E→x∈D)
*Continuing with this example, I would show how this implication (of parts i and ii) as a whole implies that A=B
But every time I attempt to do this, I feel like I go around in circles. Or at least that I'm doing too much and making it too complicated so that I forget what I'm trying to prove. please help!
The result isn't true. For example, just take .
(An explicit counterexample would be: take A= , and . Then ...)
The result might be true if you assume that for all Might that be what the OP is getting at?
**** WOW, sorry guys! BIG mistake
Asking this question might make more sense if you knew that I screwed up while asking it, of course.
What I mean to ask about is the union of these sets NOT the intersection. As in, everywhere on this problem where I put the intersection is supposed to be the union.
I actually came up with a counterexample very similar to that of Swlabr
-Thank you for your time