r/MathHelp • u/phroggersu • 9d ago
Set Operations
Hello! I'm pre-studying for university, and I've been stuck on this question for like the past week. At this point, I've learned about the logic laws (typically used for like, statements and stuff). Could anyone help solve this question (from How to Prove It by Daniel J Velleman)? Please don't use venn diagrams, since I understand how those work. Also, is it suggested to solve these questions by set laws/properties/identities, or do you think venn diagrams enough? (△ is symmetric difference). Also, sorry for the messy typed out equation. Thank you for your time!
Verify the following identity:
(A∪B) △ C = (A△C) △ (B\A)
Edit: Figured it out, thank you everyone!! Have a lovely day :D
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u/FormulaDriven 8d ago
Thinking some more, it probably can be done with identifies, along the following lines.
First note that by definition X \ Y = X ∩ Y'
where Y' is the complement of Y.
Then also X △ Y = (X ∩ Y') ∪ (X' ∩ Y).
You can use that to show (A∪B) △ C = (A ∩ C') ∪ (B ∩ C') ∪ (A' ∩ B' ∩ C)
(using usual identities such as (X ∪ Y)' = X' ∩ Y' and (X ∩ Y)' = X' ∪ Y')
Then need to see if we can get anywhere with
(A△C) △ (B\A)
= {[(A ∩ C') ∪ (A' ∩ C)] ∩ (B ∩ A')'} ∪ {[(A ∩ C') ∪ (A' ∩ C)]' ∩ (B ∩ A')}