Belongs or not?

Question

I have some collection $t_i , t_j, t_k$ . Is this conclusion true?

$ \nexists x : x\in ( (t_i \cup t_j) -( t_i \cap t_j)) $ and

$ \nexists x : x\in ( (t_i \cup t_k) -( t_i \cap t_k)) $

Then $ \nexists x : x\in ( (t_j \cup t_k) -( t_j \cap t_k)) $

I try to find counter example , but I didn't find anything , I think it's always true ,i can't proof it . is this true?

Answer 1

Your first condition is equivalent to $(A\cup B) - (A\cap B) = \emptyset$.

Since $A\cup B\supset A \supset A\cap B$, you get $$ A\cup B = A = A\cap B$$ and in the same way you get $ B = A\cap B$, hence $$ A = A\cap B = B$$ The same reasoning applies to the second condition, which yields $$ A = A\cap C = C$$ You conclude that $A=B=C$ which is equivalent to the thesis.

Answer 2

The condition $(A\cup B)-(A\cap B)=\emptyset$ is equivalent to $A=B$, which I'll let you try and prove for yourself. Thus you're asking if $t_i=t_j$ and $t_i=t_k$ implies that $t_j=t_k$, which is of course true.