Sets
Copyright © Peter Cappello 2011 Definition Visualize a dictionary as a directed graph. Nodes represent words If word w is defined in terms of word u, draw an edge from w to u. Can the dictionary be infinite? Can the dictionary have cycles? Thus, some words are not formally defined. “Set” is a primitive concept in mathematics: It is not formally defined. A set intuitively is an unordered collection of elements. Copyright © Peter Cappello 2011
Preliminaries The universe of discourse, denoted U, intuitively is a set describing the context for the duration of a discussion. E. g., let U be the set of integers. (As far as I can tell, the purpose of the Universe is to ensure that the complement of a set is a set.) Copyright © Peter Cappello 2016
Copyright © Peter Cappello 2011 Preliminaries A set S is well-defined when we can decide whether any particular object in the universe of discourse is an element of S. S is the set of all even numbers S is the set of all human beings Do we have a rule that lets us decide whether some blob of protoplasm is a human being? Copyright © Peter Cappello 2011
Definitions & Conventions A set’s objects are called its members or elements. We can describe a set with set builder notation. O = { x | x is an odd positive integer < 10 }. O = { x Z+ | x < 10 and x is odd }. By convention: N = { 0, 1, 2, 3, … } the set of natural numbers. Z = { …, -2, -1, 0, 1, 2, … } the set of integers. Z+ = { 1, 2, 3, … } the set of positive integers. Q = { p/q | p, q Z and q 0 } the set of rationals. R = the set of real numbers. Copyright © Peter Cappello 2011
Copyright © Peter Cappello 2011 Definitions Set A is a subset of set B, denoted A B, when x ( x A x B ). Set A equals set B when they have the same elements: A = B when x ( x A x B ). We can show A = B via 2 implications: A B B A x ( ( x A x B ) ( x B x A ) ). Copyright © Peter Cappello 2011
Copyright © Peter Cappello 2011 The empty set, denoted , is the set with no elements. Let A be an arbitrary set. True, false, or maybe? A. A. A A. A A. If A B A B then A is a proper subset of B, denoted A B. Copyright © Peter Cappello 2011
Copyright © Peter Cappello 2011 Venn Diagrams Venn diagram of A B. U B A Copyright © Peter Cappello 2011
Copyright © Peter Cappello 2011 Cardinality If S is a finite set with n elements, then its cardinality is n, denoted |S|. Copyright © Peter Cappello 2011
Copyright © Peter Cappello 2016 The Power Set The power set of set S, denoted P(S), is { T | T S }. What is P( { 0, 1 } )? What is P( )? What is P( P( ) )? Let P1( S ) = P( S ), Pn( S ) = P ( Pn-1( S ) ), for n > 1 | Pn ( ) | = ? Copyright © Peter Cappello 2016
Copyright © Peter Cappello 2016
Copyright © Peter Cappello 2016
Copyright © Peter Cappello 2011 | Pn ( ) | = 2| Pn-1 ( ) | Express | P5 ( ) | using only digits 2 & 0. Copyright © Peter Cappello 2011
Copyright © Peter Cappello 2011 2n is the number of vertices in a n-dimensional cube. Are there any connections between: an n-dimensional cube the power set of a set with n elements? Copyright © Peter Cappello 2011
Copyright © Peter Cappello 2011 Cartesian Products The Cartesian product of sets A and B, denoted A x B, is A x B = { ( a, b ) | a A b B }. Let S = { small, medium, large } and C = { pink, lavender }. Enumerate the ordered pairs in S x C. Enumerate the ordered pairs in C x S. Enumerate the ordered pairs in x S. | S x C | = ? Copyright © Peter Cappello 2011
Copyright © Peter Cappello 2011 Cartesian Products Generalizing Cartesian product to n sets: A1 x A2 x … x An = { ( a1, a2, …, an ) | a1 A1, a2 A2, …, an An }. Describe | A1 x A2 x … x An | in terms of the cardinalities of the component sets. Using sets S and C as previously described, describe ( S x C ) x ( C x S ). | ( S x C ) x ( C x S ) | = ? Copyright © Peter Cappello 2011
Using Set Notation with Quantifiers A shorthand for x ( x R x2 ≥ 0 ) is x R ( x2 ≥ 0 ) A shorthand for x ( x Z x2 = 1 ) is x Z ( x2 = 1 ) The statements above are either true or false. What if you want the set of elements that make a proposition function true? Copyright © Peter Cappello 2011
Truth Sets of Proposition Functions Let P be a proposition function and D a domain. The truth set of P with respect to D is { x D | P( x ) }. Enumerate the truth set { x N | ( x < 20 ) ( x is prime ) }. Copyright © Peter Cappello 2016