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Unformatted text preview: 1.5 Elementary Set Theory 11 Figure 1.3 Venn Diagrams of Different Set Operations • Commutative law for unions : A ∪ B = B ∪ A , which states that the order of the union operation on two sets is immaterial. • Commutative law for intersections: A ∩ B = B ∩ A , which states that the order of the intersection operation on two sets is immaterial. • Associative law for unions : A ∪ ( B ∪ C ) = ( A ∪ B ) ∪ C , which states that in performing the union operation on three sets, we can proceed in two ways: We can first perform the union operation on the first two sets to obtain an intermediate result and then perform the operation on the result and the third set. The same result is obtained if we first perform the operation on the last two sets and then perform the operation on the first set and the result obtained from the operation on the last two sets. • Associative law for intersections : A ∩ ( B ∩ C ) = ( A ∩ B ) ∩ C , which states that in performing the intersection operation on three sets, we can proceed in two ways: We can first perform the intersection operation on the first two sets to obtain an intermediate result and then perform the operation on the result and the third set. The same result is obtained if we first perform the operation on the last two sets and then perform the operation on the first set and the result obtained from the operation on the last two sets. • First distributive law : A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C ) , which states that the intersection of a set A and the union of two sets B and C is equal to the union of the intersection of A and B and the intersection of A and C . This law can be extended as follows: A ∩ parenleftBigg n uniondisplay i = 1 B i parenrightBigg = n uniondisplay i = 1 ( A ∩ B i ) 12 Chapter 1 Basic Probability Concepts • Second distributive law : A ∪ ( B ∩ C ) = ( A ∪ B ) ∩ ( A ∪ C ) , which states that the union of a set A and the intersection of two sets B and C is equal to the intersection of the union of A and B and the union of A and C . The law can also be extended as follows: A ∪ parenleftBigg n intersectiondisplay i = 1 B i parenrightBigg = n intersectiondisplay i = 1 ( A ∪ B i ) • De Morgan’s first law : A ∪ B = A ∩ B , which states that the complement of the union of two sets is equal to the intersection of the complements of the sets. The law can be extended to include more than two sets as follows: n uniondisplay i = 1 A i = n intersectiondisplay i = 1 A i • De Morgan’s second law : A ∩ B = A ∪ B , which states that the complement of the intersection of two sets is equal to the union of the complements of the sets. The law can also be extended to include more than two sets as follows: n intersectiondisplay i = 1 A i = n uniondisplay i = 1 A i • Other identities include the following: • A − B = A ∩ B , which states that the difference of A and B is equal to the intersection of A and the complement of B...
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 Fall '07
 Carlton
 Set Theory, Conditional Probability, Probability, basic probability concepts

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