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Survey of Mathematical Ideas Math 100 Chapter 2 John Rosson Thursday January 25, 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 Sets • A set is a collection of objects. • The objects in a set are called its elements or members. • If A is a set and a is an element of A, we show this in symbols as follows: a A a A “a is an element of A” “a is not an element of A” Designating Sets • Word description: • Listing: The set of positive whole numbers which are less than 20 and evenly divisible by 7. {7, 14} • Set builder: {x | x is a positive whole number and x is less than 20 and x is evenly divisible by 7} • The empty set, designated Ø, is the set with no elements. Set Equality Set A is equal to set B if 1. every element of A is an element of B and 2. Every element of B is an element of A. This denoted, as one would expect AB “set A equals set B” A B “set A does not equal set B” Set Equality Examples: {a} {a, a} {a, b} {b, a} {} {x | x x} " the set of even numbers" " the set of numbers divisible by 2" {x | x is a whole number and x 2 4} {x | x is a whole number and x 2} 2 Sets of Numbers. Natural or Counting numbers Whole numbers Integers Rational numbers Real numbers Irrational numbers {1,2,3,4,. ..} N {0,1,2,3,4 ,...} {...,-3,-2,-1,0,1,2,3,...} Z { p/q | p and q are integers and q 0} Q {x | x is a number tha t can be written as a decimal} R {x | x is a real number and x is not a rational number} Cardinality The cardinal number or cardinality of a set is the number of element in a set. In symbols, the cardinality of a set A is denoted n(A) Equal sets have equal cardinality, but sets with equal cardinality are not always equal. Cardinality Examples n({1,0,1}) n({1,2,3}) n({a, b, c}) 3 n() 0 n({a, b, a}) n({b, a, b}) n({a, b, a, b}) n({a, b}) 2 n({ x | x N and x is odd and x 10}) 5 Intuitively, the sets in the above example are finite. On the other hand, the sets of numbers N, Z, Q and R are all examples of infinite sets. Later, we precise definitions of the words “finite” and “infinite” mean. Subsets Set A is a subset of set B if every element of A is also an element of B Denoted in symbols, A B “set A is a subset of set B” A / B “set A is not a subset of set B” If A and B are sets then A = B if A B and B A. Subsets Set A is a proper subset of set B if A B and A≠B. This denoted in symbols, A B “set A is a proper subset of set B” A B “set A is not a proper subset of set B” Subsets Examples. Let A be a set. A A A {1,2,3} {1,2,3,4} {1,2,3,4} / {1,2,3} {1,2,3,4} {1,2,3,...} N Z Q R Sets can be elements. Any set can be an element of a set. If A {1,{1},2} then 1 2 {1} n( A) A A A 3 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 Assignments 2.3, 2.4, 2.5 Read Section 2.3 Due January 30 Exercises p. 73 1-6, 7-27, 47, 51, 52, 71, 75, 97, 115, 127, 129, 131, 133. 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.

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