Math for Elementary Teachers 0011 0010 1010 1101 0001 0100 1011 1 2 4 Chapter 2 Sets Whole Numbers, and Numeration Sets as a Basis for Whole Numbers 0011 0010 1010 1101 0001 0100 1011 • Set – a collection of objects 1. A verbal description 2. A listing of the members separated by commas or With braces {} 3. Set-builder notation 1 • 2 4 Elements(members) – objects in a set. Sets 0011 0010 1010 1101 0001 0100 1011 • Sets are denoted by capital letters – A,B,C • . indicates that an object is an element of a set • . indicates that an object is NOT an element of a set • . empty set (or null set) a set without elements. 1 2 4 Set Examples 0011 0010 1010 1101 0001 0100 1011 • Verbal – the set of states that border the Pacific Ocean • Listing A:{Alaska, California, Hawaii, Oregon, Washington} • .Oregon A • .New York A • .The Set of all States bordering Iraq . 1 2 4 More on Sets 0011 0010 1010 1101 0001 0100 1011 • Two sets are equal ( A=B) if and only if they have precisely the same elements 1 2 – Two sets, A and B, are equial if every elements of A is in B, and vice versa • If A does not equal B then A B 4 Rules regarding Sets 0011 0010 1010 1101 0001 0100 1011 1. The same element is not listed more than once within a set 2. the order of the elements in a set is immaterial. 1 2 4 One-to-One Correspondence 0011 0010 1010 1101 0001 0100 1011 • Definition: A 1-1 correspondence between two sets A and B is a pairing of the elements of A with the elements of B so that each element of A corresponds to exactly one element of B, and vice versa. If there is a 11 correspondence between sets A and B, we write A~B and say that A and B are equivalent or match. 1 2 4 One-to-One Correspondence 0011 0010 1010 1101 0001 0100 1011 • Four possible 1-1 • Equal sets are always equivalent • BUT equivalent sets are not necessarily equal • {1,2}~{a,b} BUT • {1,2} {a,b}. 1 2 4 Subset of a Set: A B 0011 0010 1010 1101 0001 0100 1011 • Definition: Set A is said to be a subset of B, if and only if every element written A B, of A is also an element of B. 1 2 4 Subset examples: 0011 0010 1010 1101 0001 0100 1011 • Vermont is a subset of the set of all New England states • {a, b, c} {a, b, c, d , e, f } • .{a, b, c} {a, b, d } 1 2 4 Subset examples continued 0011 0010 1010 1101 0001 0100 1011 • If A B and B has an element that is not in A, we write A B and say that A is a proper subset of B • Thus {a, b} {a, b, c}, since {a, b} {a, b, c} and c is in the second set but not in the first. 1 2 4 Venn Diagrams 0011 0010 1010 1101 0001 0100 1011 • U = universe • Disjoint Sets – Sets A and B have no elements in common • Sets {a,b,c} and {d,e,f} are disjoint • Sets {x,y} and {y,z} have y in common and are not disjoint. 1 2 4 Union of Sets: A B 0011 0010 1010 1101 0001 0100 1011 • Definition: The union of two sets A and B, written A B is the set that consists of all elements belonging either to a or to b (or to both). 1 2 4 Union of Sets: 0011 0010 1010 1101 0001 0100 1011 {.a, b} {c, d , e} {a, b, c, d , e} {.m, n, q} {m, n, p} {m, n, p, q} The notion of set union is the basis for the addition of whole numbers, but only when disjoint sets are used • 2+3=5 . • • • 1 2 4 Intersection of Sets: A B 0011 0010 1010 1101 0001 0100 1011 • Definition: The intersection of sets A and B, written A B is the set of all elements common to sets A and B. 1 2 4 Complement of a Set: A 0011 0010 1010 1101 0001 0100 1011 • Definition: The complement of a set A, Written A ,is the set of all elements in the universe, U, that are not in A. 1 2 4 Difference of Sets: A-B 0011 0010 1010 1101 0001 0100 1011 • Definition: The set difference (or relative complement) a set B from set A, written A-B, is the set of all elements in A that are not in B. 1 2 4 Section 2.2 0011 0010 1010 1101 0001 0100 1011 1 2 4 Whole numbers and numeration Numbers and Numerals 0011 0010 1010 1101 0001 0100 1011 • The study of the set of whole numbers W={0,1,2,3,4…} is the foundation of elementary school mathematics • A number is an idea, or an abstractions, that represents a quantity. • The symbols that we see, srite or touch when representing numbers are called numerals. 1 2 4 Three uses of whole numbers 0011 0010 1010 1101 0001 0100 1011 1. Cardinal number – whole numbers used to describe how many elements are in a finite set 2. Ordinal numbers - concerned with order e.g. your team is in fourth place 3. Identification numbers – used to name things – credit card, telephone number, etc it’s a symbol for something. 1 2 4 0011 0010 1010 1101 0001 0100 1011 • The symbol n(A) is used to represent the number of elements in a finite set A. • n({a,b,c})=3 • n({a,b,c,…,z})=26. 1 2 4 Ordering Whole Numbers (1-1 correspondences) 0011 0010 1010 1101 0001 0100 1011 • Definition: Ordering Whole Numbers: 2 – Let a=n(A) and b=n(B) then a<b (read a is less than b) or b>a (b is greater than a) if A is equivalent to a proper subset of B. 1 4 Problem: determine which is greater 3 or 8 in three different ways 0011 0010 1010 1101 0001 0100 1011 1. Counting chant – one, two, three, etc 2. Set Method – a set with three elements can be matched with a proper subset of a set with eight elements 3<8 and 8>3. 1 2 4 Problem: determine which is greater 3 or 8 in three different ways (cont) 0011 0010 1010 1101 0001 0100 1011 3. Whole-Number Line – since 3 is to the left of 8 on the number line, 3 is less than 8 and 8 is greater than 3. 1 2 4 Numeration Systems 0011 0010 1010 1101 0001 0100 1011 • Tally numeration system – single strokes, one for each object counted. 1 • Improved with grouping. 2 4 The Egyptian Numeration System 0011 0010 1010 1101 0001 0100 1011 • developed around 3400 B.C invovles grouping by ten. 1 • • =? 321. 2 4 The Roman Numeration System 0011 0010 1010 1101 0001 0100 1011 • Developed between 500 B.C. and A.D. 100 • The values are found by adding the values of the various basic numerals • MCVIII is 1000+100+5+1+1+1=1108 • New elements 1 – Subtractive principle – Multiplicative principle. 2 4 Subtractive system 0011 0010 1010 1101 0001 0100 1011 • Permits simplifications using combinations of basic numbers • IV – take one from five instead of IIII • The value of the pair is the value of the larger less the value of the smaller. 1 2 4 Multiplicative System 0011 0010 1010 1101 0001 0100 1011 • Utilizes a horizontal bar above a numeral to represent 1000 times the number • Then V means 5 times 1000 or 5000 • and XI is 1100 • System still needs many more symbols than current system and is cumbersome for doing arithmetic. 1 2 4 The Babylonian Numeration System 0011 0010 1010 1101 0001 0100 1011 • Evolved between 3000 and 2000 B.C. • Used only two numerals, one and ten 1 2 4 • for numbers up to 59 system was simply additive • Introduced the notion of place value – symbols have different values depending on the place they are written. Sections 2.3 The Hindu-Arabic System 0011 0010 1010 1101 0001 0100 1011 1. Digits 0,1,2,3,4,5,6,7,8,9 – 10 digits can be used in combination to represent all possible numbers 2. Grouping by tens (decimal system) known as the base of the system – Arabic is a base ten system 3. Place value (positional) Each of the various places in the number has it’s own value. 1 2 4 Models for multi digit numbers 0011 0010 1010 1101 0001 0100 1011 • Bundles of Sticks – each ten sticks bound together with a band 1 2 • Base ten pieces (Dienes blocks) individual cubes grouped in tens. 4 The Hindu-Arabic System 0011 0010 1010 1101 0001 0100 1011 4. Additive and multiplicative • The value of a Hindu-Arabic numeral is found by multiplying each place value by its corresponding digit and then adding all of the resulting products. Place values: thousand hundred ten one Digits 6 5 2 3 Numeral value 6x1000 + 5x100 + 2x10 + 3x1 Numeral 6523. 1 2 4 Observations about the naming procedure 0011 0010 1010 1101 0001 0100 1011 1. 2. 3. 4. 5. The number 0,1,…12 all have unique names The numbers 13,14, …19 are the “teens” The numbers 20,…99 are combinations of earlier names but reversed from the teens in that the tens place is named first e.g. 57 is “fifty-seven The number 100, … 999 are combinations of hundreds and previous names e.g. 637 reads “six hundred thirtyseven” In numerals containing more than three digits, groups of three digits are usually set off by commas e.g. 123,456,789 . 1 2 4 Learning 0011 0010 1010 1101 0001 0100 1011 • Three distinct ideas that children need to learn to understand the Hindu-Arabic numeration system . 1 2 4 Base 5 operations 0011 0010 1010 1101 0001 0100 1011 • We can express numeration systems as base systems 1 2 – The number 18 in Hindu-Arabic can be stated as 18ten 18 base ten – To study a system with only five digits (0,1,2,3,4) we would call that a base 5 system e.g. base five 37five . 4