7 - ) = ( A ∪ B ) ∩ ( A ∪ C ) A ∪ ( A ∩ B ) = A A...

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For example, let A = { 1 , 3 , 5 , 7 , 9 } and B = { 3 , 5 , 6 , 10 , 11 } . Then A B = { 1 , 3 , 5 , 6 , 7 , 9 , 10 , 11 } A B = { 3 , 5 } A - B = { 1 , 7 , 9 } A ± B = { 1 , 7 , 9 , 6 , 10 , 11 } It is often helpful to illustrate these combinations of sets using Venn dia- grams 2 . 2.2.2 Properties of Operators In this section, we investigate certain equalities between sets constructed from our set-theoretic operations. P ROPOSITION 2.6 (P ROPERTIES OF OPERATORS ) Let A , B and C be arbitrary sets. They satisfy the following properties: Commutativity Idempotence A B = B A A A = A A B = B A A A = A Associativity Empty set A ( B C ) = ( A B ) C A ∪ ∅ = A A ( B C ) = ( A B ) C A ∩ ∅ = Distributivity Absorption A ( B C
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Unformatted text preview: ) = ( A ∪ B ) ∩ ( A ∪ C ) A ∪ ( A ∩ B ) = A A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C ) A ∩ ( A ∪ B ) = A Proof We will just look at the ±rst distributivity equality. Some of the other cases will be set as exercises. Draw a Venn diagram to give some evidence that the property is indeed true. It is not a proof! Let A , B and C be 2 I will check whether you have been taught Venn diagrams. If not, we will go over them during lectures. 8...
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This note was uploaded on 01/02/2010 for the course MATH Math2009 taught by Professor Koskesh during the Spring '09 term at SUNY Empire State.

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