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Unformatted text preview: B ⊆ C … BUT such an x may not exist (if A = φ ) Proving two sets are equal In general to show that X = Y we need to show two things: X ⊆ Y and Y ⊆ X Occasionally, however, the proof follows directly by logical equivalence: Example (DeMorgan’s Law): A ∪ B = A ∩ B Proof : A ∪ B = { x ∈ U x ∉ A ∪ B } = { x ∈ U ¬ ( x ∈ A ∪ B ) } = { x ∈ U ¬ ( x ∈ A ∨ x ∈ B ) } = { x ∈ U ¬ ( x ∈ A ) ∧ ¬ ( x ∈ B ) } = { x ∈ U ( x ∉ A ) ∧ ( x ∉ B ) } = { x ∈ U ( x ∈ A ) ∧ ( x ∈ B ) } = A ∩ B...
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 Spring '10
 fleck
 Logic, Philosophy of mathematics, Transitive relation, Proof theory

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