am_lect_13 - Proving two sets are equal In general to show...

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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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Claim : A = ( A B ) ( A B ) Proof : Let x A There are two cases: x B and x B If x B , then ( x A ) ( x B ) and hence x A B Thus x ( A B ) ( A B ) If x B , then ( x A ) ( x B ) and hence x ( A B ) Thus x ( A B ) ( A B ) We have shown that A ( A B ) ( A B ) Now suppose x ( A B ) ( A B ), so ( x A B ) or ( x A
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am_lect_13 - Proving two sets are equal In general to show...

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