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#1 2011-03-04 01:25:35

Kryptonis
Member
Registered: 2011-03-03
Posts: 11

set help

Let A, B and C be arbitrary sets taken from the positive integers.
    Prove the following statement: If A − B ⊆ C , then A ⊆ B ∪C

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#2 2011-03-04 02:13:14

Kryptonis
Member
Registered: 2011-03-03
Posts: 11

Re: set help

Figured it out...
    A − B ⊆ C ≡ (A ∩ ¬B) ⊆ C                Def of Diff
    ≡ {x | (x ∈ A ∧ x  ¬∈ B) → x ∈ C}            Def of Diff, Def of subset
    ≡ {x | (x ¬∈  A ∨ x ∈ B) ∨ x ∈ C}            Log Equiv, De Morg
    ≡ {x | (x ∈ A → x ∈ B) ∪ x ∈ C}            Log Equiv, Def of Union
    ≡ A ⊆ B ∪C                                Set Builder Notation

Last edited by Kryptonis (2011-03-04 02:14:51)

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