09-01-2009Sets, Counting and The Whole NumbersQuestion? Why do we have numbers? How do you think they were invented? What different ways arenumbers used?Numbers. Three types:• nominal or identification: A number used as a label. Ex.: Phone numbers, student ID numbers.• An ordinal number is a number from a list of ordered numbers. Ex.: raffle ticket numbers, year2009.• A cardinal number measures the number of objects in a set. Ex.: Fifteen lottery tickets. 32books on the shelf.One-to-one correspondence of sets and equivalent sets.• Recall, two sets are equal if they contain exactly the same elements. Ex. If A = {x, y, z} andB = {z, y, x} then A equals B and we write A = B.• We say that there is a one-to-one correspondence between the sets A and B if every element ofthe set A can be paired off with exactly one element of B so that every element of A has exactlyone partner from B and every element of B has exactly one partner from A.• More mathematically, we say there is a one-to-one correspondence between the sets A and B ifthere is a way to assign exactly one element of B to every element of A which uses all the elementsof B exactly one time.• When there is a one-to-one correspondence between the sets A and B we say the sets are equiva-lent. We write A ∼ B.• Examples.: C = { Barack Obama, Joe Biden, Hillary Clinton } and P = { president, vice-president,secretary of state }. A = {a, b, c, d} and N4= {1, 2, 3, 4}.• Note that the examples show that A ∼ P is not the same as A = B. So being equivalent is not thesame as being equal. However if A = B then certainly A ∼ B.The whole numbers and the size of a set.• Recall thatN = {1, 2, 3, 4, ...}is the set of natural or counting numbers. How many elements are in this set?• We say a set is finite if it is equivalent to either ∅ or to the set {1, 2, 3, 4, ..., n} where n is a naturalnumber.• If a set is not finite it is called infinite. Ex.: Odd numbers, prime numbers.• We will write n(A) to denote the number of elements in the set A. The number of elements in aset is called the cardinality or cardinal number of the set.• The set W is the set set of whole numbers. This is the set of numbers which could be the cardinalnumber of a set. So of courseW = {0, 1, 2, 3, ...}.• Examples.: Find (a) the cardinality of the set of days in the week, (b) n(E) where E is the set ofeven numbers (c) n(∅ ∩W ) (d) cardinal number of {0} and (e) n(A) where A = {x ∈ W |x2< 100}.Models of Whole Numbers. Tiles, blocks, strips, number line where whole numbers represent thedistance from 0. Page 95.Ordering the whole numbers.• Recall the definition of a subset: A ⊂ B if every element of A is also an element of B.• We will say A is a proper subset of B if A ⊂ B and A 6= B.• Now let a = n(A) and b = N (B) be whole numbers where A and B are finite sets. If A is equivalentto a proper subset of B then a is less than b and we can write a < b.• Ex.: Demonstrate that 2 < 5 with models: objects, tiles, strips, number lines.More problems with Venn-diagrams. Use Venn diagrams to solve the following problems:• There are 150 senators on the student senate and there are three optional committees: Greek life(G), academic affairs (A), and social life (S). There are 30 senators on G, 50 on A, and 40 on S.Also, 10 senators are on A ∩ G, 10 are on A ∩ S, 15 are on S ∩ G. There are 5 senators on all threecommittees. How many senators did not volunteer for any committees?• There are 25 people at the ice cream social. 15 have chocolate; 15 have vanilla, 10 have strawberry;10 have both chocolate and vanilla; 5 have both chocolate and strawberry; 5 have both strawberryand vanilla; 3 people have all three. How many people have no ice-cream?
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