key2fa00 - COT3100C-01, Fall 2000 S. Lang Solution Key to...

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Unformatted text preview: COT3100C-01, Fall 2000 S. Lang Solution Key to Assignment #2 (40 pts.) 9/26/2000 1. (6 pts.) The following Venn diagram shows three sets A , B , and C contained within a universe set named U . In the diagram, different subsets (regions) are labeled by the numbers 1 through 8. Thus, set A contains subsets labeled 1, 2, 4, and 5; set B contains subsets labeled 2, 3, 5, and 6; set C contains subsets labeled 4, 5, 6, and 7; set U contains subsets labeled 1 through 8 (i.e., the universe). Now, give the labels (i.e., numbers) of the subsets that are contained in each of the following sets: 2. (24 pts.) Prove each of the following statements (a) – (f), assuming the symbols A , B , and C represent sets. You are allowed to use appropriate definitions, and the following theorems and laws ( but only these ) in the proof. Be sure to explain each step of your proof. (Commutative Law) A ∪ B = B ∪ A , A ∩ B = B ∩ A . (Associative Law) ( A ∪ B ) ∪ C = A ∪ ( B ∪ C ), ( A ∩ B ) ∩ C = A ∩ ( B ∩ C ). (Distributive Law) A ∪ ( B ∩ C ) = ( A ∪ B ) ∩ ( A ∪ C ) , A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C ) . (Idempotent Property) A ∪ A = A , A ∩ A = A . (De Morgan’s Law) ¬ ( A ∪ B ) = ¬ A ∩ ¬ B , ¬ ( A ∩ B ) = ¬ A ∪ ¬ B . (Double Negation) ¬ ( ¬ A ) = A . (Complementary Property) A ∩ ¬ A = ∅ , A ∪ ¬ A = U , where U denotes the universe....
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key2fa00 - COT3100C-01, Fall 2000 S. Lang Solution Key to...

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