Sets
Copyright © Peter Cappello 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 Preliminaries The universe of discourse, denoted U, intuitively is a set describing the context for the duration of a discussion. E. g., U is the set of integers. (As far as I can tell, its purpose is to ensure that the complement of a set is a set.)
Copyright © Peter Cappello 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 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 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 The empty set, denoted , is the set with no elements. Let A be an arbitrary set. True, false, or maybe? 1. A. 2. A. 3. A A. 4. A A. If A B A B then A is a proper subset of B, denoted A B.
Copyright © Peter Cappello Venn Diagrams Venn diagram of A B. U A B
Copyright © Peter Cappello Cardinality If S is a finite set with n elements, then its cardinality is n, denoted |S|.
Copyright © Peter Cappello 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 P 1 ( S ) = P( S ), P n ( S ) = P ( P n-1 ( S ) ). | P n ( ) | = ?
P 1 ( ) = { } P 2 ( ) = { , { } } P 3 ( ) = { , { }, {{ }}, { , { } } } | P 1 ( ) | = 2 0 | P n ( ) | = 2 | Pn-1 ( ) | Express | P 5 ( ) | using only digits 2 & 0. Copyright © Peter Cappello
Copyright © Peter Cappello 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 Cartesian Products Generalizing Cartesian product to n sets: A 1 x A 2 x … x A n = { ( a 1, a 2, …, a n ) | a 1 A 1, a 2 A 2, …, a n A n }. Describe | A 1 x A 2 x … x A n | 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 Using Set Notation with Quantifiers A shorthand for x ( x R x 2 ≥ 0 ) is x R ( x 2 ≥ 0 ) A shorthand for x ( x Z x 2 = 1 ) is x Z ( x 2 = 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 Truth Sets of Proposition Functions Let P be a proposition function, 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 ) }.