slides07 - Set Theory CMSC 250 1 Set definitions Definition...

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1 CMSC 250 Set Theory
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2 CMSC 250 Set definitions Definition: A set is a collection of objects. Examples: A = {1,2,3} B = { x Z | - 4 < x < 4} C = { x Z + | - 4 < x < 4} A set is completely defined by its elements, i.e., { a , b } = { b , a } = { a , b , a } = { a , a , a , b , b , b }
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3 CMSC 250 More set concepts The universal set U is the set of all elements under consideration A set can be finite or can be infinite For a set S , n( S ) or | S | are used to refer to the cardinality of S , which is the number of elements in S The symbol means "is an element of" The symbol means "is not an element of"
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4 CMSC 250 Subset A B ( 2200 x U )[ x A x B ] A is contained in B B contains A A B ( 5 x U )[ x A x B ] Relationship between membership and subset: ( 2200 x U )[ x A { x } A ] Definition of set equality: A = B A B B A
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5 CMSC 250 Proper subset B A B A B A
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6 CMSC 250 Do these represent the same sets or not? X = { x Z | ( 5 p Z )[ x = 2 p ]} Y = { y Z | ( 5 q Z )[ y = 2 q - 2]} A = { x Z | ( 5 i Z )[ x = 2 i + 1]} B = { x Z | ( 5 i Z )[ x = 3 i + 1]} C = { x Z | ( 5 i Z )[ x = 4 i + 1]}
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7 CMSC 250 Formal definitions of set operations Union: Intersection: Complement: Difference: } { B x A x B A = } { B x A x B A = } { B x A x B A = - } | { ' A x U x A A A c = = = B A B A = -
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8 CMSC 250 Venn Diagrams Sets are represented as regions to illustrate relationships between them.
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