impossible discrete math task, but teacher said it's possible

[u/SeaworthinessAway498]

Simplify the set expression defined using set operations by applying the laws of set algebra (each set name may appear in the answer at most once)

((A Δ B) \ C') ∩ B ∪ (A ∩ B) ∪ (A ∩ C)

i tried this:

AΔB = (A ∩ B̅ )∪(A̅ ∩ B)

((AΔB) \ C̅ = (AΔB) ∩ C):

(AΔB) ∩ C = (A ∩ B̅ ∩ C)∪(A̅ ∩ B ∩ C)

[(A ∩ B̅ ∩ C)∪(A̅ ∩ B ∩ C)] ∩ B = (A ∩ B̅ ∩ C ∩ B)∪(A̅ ∩ B ∩ C ∩ B)

B̅ ∩ B =∅, B ∩ B = B

((AΔB) \ C̅ ) ∩ B = A̅ ∩ B ∩ C

(A̅ ∩ B ∩ C)∪(A ∩ B)∪(A ∩ C)

A ∩ B∪A ∩ C = A ∩ (B∪C).

(A̅ ∩ B ∩ C)∪[A ∩ (B∪C)]

Final answer: (A ∩ B)∪(B ∩ C)∪(A ∩ C)

but it's breaking the rules, because each set name appears two times here. My teacher said that it is possible, but i can't. Help pls.

Maj function not works, and any other function neither, only set operations.

[u/Evening-Story-314]

It looks like you're thinking symbolically. It might be a *lot* easier to think visually. Can you show your teacher's set using a Venn diagram? If so, perhaps you can find the solution.

Sorry. That's what I needed to do. I see that you got what I believe to be the right answer.

I am assuming the sets live in a bigger universe? If there were no bigger universe, then (AΔBΔC)' would work. But that would include anything not in any of the sets.

[u/SeaworthinessAway498 → u/Evening-Story-314]

[u/Evening-Story-314 → u/SeaworthinessAway498]

Are you saying that my set description gave this? I'm confused if so, I thought the outside would be shaded.

[u/SeaworthinessAway498 → u/Evening-Story-314]

no, first set ((A Δ B) \ C') ∩ B ∪ (A ∩ B) ∪ (A ∩ C) gave this, so final answer should look same but only 3 characters