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Unformatted text preview: A * C does not logically follow from A ⊆ B,B * C . A counterexample: A = { a } B = { a,b } C = { a } The premises are true, but the conclusion is false. A smaller counterexample is: A = ∅ ,B = { a } ,C = ∅ 3. Given B ⊆ A , show A ∩ B = ∅ using natural deduction. 1. B ⊆ A premise 2. ∀ x • x ∈ B ⇒ x ∈ A 1 , subset 3. ∀ x • x ∈ B ⇒ ( x ∈ U ∧ x 6∈ A ) 2 , complement 4. z g 5. z g ∈ ( A ∩ B ) assumption 6. z g ∈ A ∧ z g ∈ B 5 , set intersection 7. z g ∈ A 6 , ∧ E 8. z g ∈ B 6 , ∧ E 9. z g ∈ B ⇒ ( z g ∈ U ∧ z g 6∈ A ) 3 , ∀ E 10. z g ∈ U ∧ z g 6∈ A 8 , 9 , ⇒ E 11. z g 6∈ A 10 , ∧ E 12. false 7 , 11 , ¬ E 13. ¬ ( z g ∈ ( A ∩ B )) 512 , ¬ I 14. ∀ z • ¬ ( z ∈ ( A ∩ B )) 413 , ∀ I 15. A ∩ B = ∅ 14 , empty set 2...
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 Spring '08
 WORMER
 Computer Science, Set Theory, Naive set theory, Empty set, Intersection, University of Waterloo School of Computer Science

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