Symmetric difference of set identity

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SUMMARY

The discussion focuses on verifying the identity of symmetric difference in set theory without relying on Venn diagrams or truth tables. Participants emphasize the associative and commutative properties of sets when applying the symmetric difference operator. By combining like terms and simplifying, users can create shortcuts for proving variations of this identity. This formal approach enhances the understanding and application of set identities in mathematical proofs.

PREREQUISITES
  • Understanding of set theory concepts, particularly symmetric difference.
  • Familiarity with the associative and commutative properties of operations.
  • Basic knowledge of formal mathematical proofs.
  • Experience with simplifying algebraic expressions involving sets.
NEXT STEPS
  • Research the properties of symmetric difference in set theory.
  • Explore techniques for simplifying set identities.
  • Learn about constructing formal proofs in set theory.
  • Investigate variations of set identities and their applications in proofs.
USEFUL FOR

Mathematicians, educators, and students studying set theory, particularly those interested in formal proofs and identity verification using symmetric difference.

el_llavero
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Is there a shorter way to verify this identity, as you can see I haven't even finished it. I know you can use Ven diagrams and truth tables but I wanted to avoid them inorder to use a more general formal approach. picture is attached
 

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Sets are associative and commutative over the symmetric difference operator, use this to combine like terms, then simplify. Similar technique can be used on variations of this identity to build your own identities which can be used as short cuts in other proofs containing the symmetric difference operator.

see attachment
 

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