©1999 indiana university trustees basic set theory definitions a set is a collection of objects or...
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©1999 Indiana University Trustees
Basic Set Theory DefinitionsBasic Set Theory Definitions
A set is a collection of objects or elements
An element is an object that make up a set
The universal set (parent set) contains ALL of the elements considered in a given problem; serves as an encompassing set for various subsets. (contains various subsets)
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Simple Set ExampleSimple Set Example the universal set is
a deck of ordinary playing cards
each card is an element in the universal set
some subsets are:– face cards– numbered cards– suits– poker hands
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Universal SetsUniversal Sets
The universal set is the set of all things pertinent to to a given discussionand is designated by the symbol U
Example:U = {all students at IUPUI}Some Subsets:
A = {all Computer Technology students}B = {freshmen students}C = {sophomore students}
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SubsetsSubsets
a subset whose elements are all members of another set
notation:
means “is a subset of”
means “is a proper subset of”
means “is not a subset of”
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The Empty SetThe Empty Set
empty set (null set) - any set that contains no elements (the empty set is a subset of every set including itself)
Union - the set of all the elements in two or more given sets.
Intersection – the set of all the elements that are common to two or more given sets.
Disjoint set – a set with no elements in common.
Complement – the set of all elements in the universal set that are not in a given set, which is a subset of the universal set.
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©1999 Indiana University Trustees
Venn DiagramsVenn Diagrams
Venn diagrams show relationships between sets and their elements
Universal Set
Sets A & B
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Venn Diagram Example 1Venn Diagram Example 1
Set Definition ElementsA = {x | x Z+ and x 8} 1 2 3 4 5 6
7 8B = {x | x Z+; x is even and 10} 2 4
6 8 10A BB A
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Venn Diagram Example 2Venn Diagram Example 2
Set Definition ElementsA = {x | x Z+ and x 9} 1 2 3 4 5
6 7 8 9B = {x | x Z+ ; x is even and 8} 2 4
6 8
A BB AA B
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Venn Diagram Example 3Venn Diagram Example 3
Set Definition ElementsA = {x | x Z+ ; x is even and 10} 2 4
6 8 10B = x Z+ ; x is odd and x 10 } 1 3
5 7 9
A BB A
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Venn Diagram Example 4Venn Diagram Example 4Set Definition
U = {1, 2, 3, 4, 5, 6, 7, 8}
A = {1, 2, 6, 7}B = {2, 3, 4, 7}C = {4, 5, 6, 7}
A = {1, 2, 6, 7}
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Venn Diagram Example 5Venn Diagram Example 5
Set Definition U = {1, 2, 3, 4, 5, 6, 7, 8}
A = {1, 2, 6, 7}B = {2, 3, 4, 7}C = {4, 5, 6, 7}
B = {2, 3, 4, 7}
©1999 Indiana University Trustees
Venn Diagram Example 6Venn Diagram Example 6
Set Definition U = {1, 2, 3, 4, 5, 6, 7, 8}
A = {1, 2, 6, 7}B = {2, 3, 4, 7}C = {4, 5, 6, 7}
C = {4, 5, 6, 7}