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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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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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