COSI 155b Chapter Notes - Chapter 2: Well-Ordering Principle
5 Jun 2018
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Cartesian Products
Set – Builder Notation
• Q = {…, (1/1), (1/2), (1/3) ,… } = {(m/n)|m, n € Z, n ≠ 0}
• 2Z+1 = {…, -3, -1, 1, 3, …} = {2n+1|n € Z}
Ordered Pairs
• An ordered pair (a, b) is a set: {{a}, {a, b}}
• The Cartesian Product, AxB, is the set: {(a, b)| a € A and b € B}.
• Ex. Given a = {0, 1, 2} and b = {0, 1},
o axb = {(0,0),(0,1),(1,0),(1,1),(2,0),(2,1)}
o bxa = {(0,0),(0,1),(0,2),(1,0),(1,1),(1,2)}
Cardinality of Products
• The cardinality of AxB is the cardinality of A multiplied by the cardinality of B.
Cartesian Products and Sets
• Cartesian Products also work with more than two sets.
• Ex. AxBxC
Products and the Empty Set
• Given A = {1, 2, 3}, {}xA = {}.
Well Ordering Principle
• Any non-empty subset of N has a “least” element.
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Indexed Sets
• Unions and Intersections can be shortened via indexed sets.
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