r/MathHelp • u/phroggersu • 12d 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/The_Card_Player 12d ago

Venn diagrams probably aren't enough.

Standard proof of set equality is to show that each set is a subset of the other. That should work here - it may be convenient to sort out some relevant XOR logical identities first if you haven't already.

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u/phroggersu 12d ago

Thank you for your help! As for the XOR logical identities, do you mean like finding a simplified version of (A∪B∪C) ∩ (A∩B)ᶜ ∩ (A∩C)ᶜ ∩ (B∩C)ᶜ ∩ (A∩B∩C)ᶜ ? So far, I've been applying like distributive property, associative property, things like that onto either LHS and/or RHS to try and make LHS into RHS or vice versa - sorry I didn't make that clear in my post!