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Kryptonis

Member

Registered: 2011-03-03

Posts: 11

Set operations, proof

Let A, B, and C be sets such that C  ⊂ B (i.e., C is a proper subset of B, or possibly C = B). Use appropriate set theoretic laws and theorems to prove     that (A – B) ∪ (B – C) = ¬C ∩ (A ∪ B). Be sure to explain each step of your proof.

This is what i have, and i have tried several ways just can't quite seem to get it right...    Any help would be great, ty in advance!

(A – B) ∪ (B – C)
    ≡{x | x ∈ A ∧ x   B}∪{x | x ∈ B ∧ x   C}            Def of diff
    ≡{x | (x ∈ A ∧ x   B) ∨ ( x ∈ B ∧ x   C)}            Def of union
    ≡{x | (x ∈ A ∧ x   B) ∨ ( x   B ∨ x ∈ C)}            De Morgan
    ≡{x | x ∈ A ∧ (x   B ∨ x ∈ C)}                       Idem & Assoc
    ≡{x | x ∈ A ∧ (x ∈ B ∧ x   C)}                    De Morgan
    ≡{x | x   C x ∧ (∈ A ∧ x ∈ B)}                    Assoc
    ≡{x | x   C x ∩ (∈ A ∩ x ∈ B)}                    Def of Inter
    ≡ ¬C ∩ (A ∩ B)                        Def of Set Build Notation

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Set operations, proof

Hi Kryptonis,

---

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."