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!