1. Sets

2. Copyright © Peter Cappello 20112 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.

3. Copyright © Peter Cappello 20113 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.)

4. Copyright © Peter Cappello 20114 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?

5. Copyright © Peter Cappello 20115 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.

6. Copyright © Peter Cappello 20116 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 ) ).

7. Copyright © Peter Cappello 20117 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.

8. Copyright © Peter Cappello 20118 Venn Diagrams Venn diagram of A  B. U A B

9. Copyright © Peter Cappello 20119 Cardinality If S is a finite set with n elements, then its cardinality is n, denoted |S|.

10. Copyright © Peter Cappello 201110 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 (  ) | = ?

11. 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 201111

12. Copyright © Peter Cappello 201112 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 | = ?

13. Copyright © Peter Cappello 201113 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 ) | = ?

14. Copyright © Peter Cappello 201114 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?

15. Copyright © Peter Cappello 201115 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 ) }.