Identity Proof: (A-B)-C=A-(B∪C)

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    Identity Proof Set
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SUMMARY

The identity proof for the equation (A-B)-C=A-(B∪C) is established using set identities. Specifically, the identities A-B=A\bar{B} and $\overline{B\cup C}=\bar{B}\bar{C}$ are crucial for the proof. The discussion emphasizes the importance of understanding set operations, including the complement and union of sets, to validate the equality. Additional set identities, such as commutativity and associativity, are acknowledged as relevant but not necessary for this specific proof.

PREREQUISITES
  • Understanding of set theory, including operations like union and intersection.
  • Familiarity with set identities, particularly A-B=A\bar{B} and $\overline{B\cup C}=\bar{B}\bar{C}$.
  • Knowledge of set complements and their notation.
  • Basic mathematical proof techniques, especially in the context of set operations.
NEXT STEPS
  • Study the properties of set complements in detail.
  • Learn about additional set identities, including commutativity and associativity of union and intersection.
  • Explore advanced topics in set theory, such as De Morgan's laws.
  • Practice proving set identities through various examples and exercises.
USEFUL FOR

Mathematicians, students studying set theory, educators teaching mathematical proofs, and anyone interested in formal logic and set operations.

gicm
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Show that(A-B)-C=A-(BUC)
 
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This can be shown using identities on sets. Two identities are: $A-B=A\bar{B}$ and $\overline{B\cup C}=\bar{B}\bar{C}$. Here $\bar{A}$ denotes the complement of $A$, and I skip intersection, i.e., I write $AB$ for $A\cap B$. There are numerous other identities on sets, such as commutativity and associativity of intersection and union, laws involving the empty set and so on. Can you use the ones I provided to prove your equality?
 

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