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Survey of Mathematical Ideas Math 100 Chapter 2 John Rosson Tuesday January 30, 2007 Basic Concepts of Set Theory 1. 2. 3. 4. 5. Symbols and Terminology Venn Diagrams and Subsets Set Operations and Cartesian Products Cardinal Numbers and Surveys Infinite Sets and Their Cardinalities Power Set The power set of set A, denoted P( A) is the set of all subsets of A. Thus P(A) {x | x A} Power Set Example P({1,2,3}) { , {1},{2},{3}, {1,2},{1,3},{2,3}, {1,2,3} } In particular, the number of subsets of {1,2,3} is n(P({1,2,3})) 8 Power Set Theorem: The number of subsets of a finite set A is given by n(P( A)) 2n( A) and the number of proper subsets is given by 2 n( A) 1 Power Set Set Cardinality # Subsets # Proper Subsets Ø 0 20=1 1-1=0 {a} 1 21=2 2-1=1 {a,b} 2 22=4 4-1=3 {1,2,3} 3 23=8 7 {1,2,c,4,5} {1,2,3,…,100} 5 100 25=32 31 2100=12676 12676506002 5060022822 28229401496 9401496703 703205375 205376 Complement The collection of all possible element of sets, either stated or implied, is called the universal set, often denoted U. For any subset A of the universal set U, the complement of A, denoted A is the set elements of U not in A. Thus A {x | x U and x A} Complement Examples. Let U={a,b,c,d,e,f,g,h,i,j,k,l}, A={a,b,c,d}. Let C D. A {e, f , g , h, i, j , k , l} ( A) {e, f , g , h, i, j , k , l} A {a} {b, c, d , e, f , g , h, i, j , k , l} U U D C Venn Diagrams A Venn diagram is a pictorial representation of sets and their various relations and operation. The first picture below represents the universal set U, a set A, and the complement of A. The second represents the relation M N Numbers as Sets All mathematical objects can be defined in terms of sets. The example below indicates how one might define the first five whole numbers as sets. 0 1 {0} {} 2 {0,1} {,{}} 3 {0,1,2} {,{},{,{}}} 4 {0,1,2,3} {,{},{,{}},{,{},{,{}}}} Intersection The intersection of sets A and B, denoted A B is the set of elements common to both A and B. A B {x | x A and x B} Intersection Let A and B be sets with A B. Let C = {1,2,3} and D = {3,a,b}. Let U be the universal set A U A A A A A B A C D {1,2,3} {3, a, b} {3} Sets with empty intersection are called disjoint. Thus, every set is disjoint from its complement. Union The union of sets A and B, denoted A B is the set of elements belonging to either of the sets. A B {x | x A or x B} Union Let A and B be sets with A B. Let C = {1,2,3} and D = {3,a,b}. Let U be the universal set A A U AU A A U A B B C D {1,2,3} {3, a, b} {1,2,3, a, b} Difference The difference of sets A and B, denoted A B is the set of elements belonging to set A but not to set B. A B {x | x A and x B} Difference Let A and B be sets with A B. Let C = {1,2,3} and D = {3,a,b}. Let U be the universal set A A A A A U A A A U A A A C D {1,2,3} {3, a, b} {1,2} D C {3, a, b} {1,2,3} {a, b} Ordered Pairs The ordered pair of with first component a and second component b, denoted ( a, b) is defined to be the set (a, b) {{a},{a, b}}. Thus, (a, b) (c, d ) only if a c and b d . Note that for ordered pairs, order is important. So (a, b) (b, a) only if a b. In particular, (a, a) {a, a} Cartesian Product The Cartesian product of sets A and B, denoted A B is the set A B {( a, b) | a A and b B}. Cartesian Product Let C = {1,2,3} and D = {3,a,b}. C D {(1,3), (1, a), (1, b), (2,3), (2, a), (2, b), (3,3), (3, a ), (3, b)} In general, for sets A and B: n( A B) n( A) n( B). So in the example n(C D) n(C ) n( D) 3 3 9. Operations on Sets Operations on sets can be combined. Let A={a,b}, B={b,c}, C={c,d}, D={b,d} and E={a,c}. Calculate A B C D E in list form. Working from the inside out A B C D E {a,b} {b,c} C D E {b} {c,d} D E {b,c,d} {b,d} E {c} {a,c} {(c,a),(c,c)} Venn Diagrams Here is the previous set calculation as a Venn diagram. The is no adequate Venn diagram for the Cartesian product. A A B b b c a B ( A B) C b b d c d c C D (( A B) C ) D b d c c De Morgan’s Laws For and sets A and B, the complement of their intersection is the union of their complements, ( A B) A B and the complement of their unions is the intersection of their complements. (A B) A B De Morgan’s Laws The set A B will be all the blue not in A and not in B. Assignments 2.4, 2.5, 3.1 Read Section 2.4 Due February 1 Exercises p. 79 1, 3, 5, 7, 9, 17, 19, 25, and 27. Read Section 2.5 Due February 6 Exercises p. 88 1-6, 7, 9, 11, 13, 14, 15, 24, 29, 32, 37, 38, 39, 40, 43. Read Section 3.1 Due February 8 Exercises p. 99 1-9, 39-47, 49-53, 57-74