CHAPTER 2. Set, Whole Numbers, and Numeration

Transcription

1 CHAPTER 2 Set, Whole Numbers, and Numeration 2.1. Sets as a Basis for Whole Numbers A set is a collection of objects, called the elements or members of the set. Three common ways to define sets: (1) A verbal description. The set of all sophomore female students in this class This is a clear criterion for judging set membership. (2) Listing (within braces): {orange, pink, green} (3) Set-builder notation: Note. (1) If S = {2, 4, 6, 8, 10}, {x x is an even whole number} = {2, 4, 6, 8,... } 4 2 S (4 is an element of S) and 7 /2 S (7 is not an element of S). (2) ; (preferably) or { } denotes the empty set (or null set), the set with no members. 11

2 12 2. SET, WHOLE NUMBERS, AND NUMERATION Two sets A and B are equal (A = B) if they have precisely the same elements. Note. (1) We write A 6= B if A and B are not equal. (2) Elements are usually listed only once in a set, so {a, a, b, c} = {a, b, c}. (3) The order of listing is immaterial, so (4) {;} 6= ; (or {{ }} 6= { }). Why? {a, b, c} = {c, a, b}. 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 with exactly one element of B, and vice-versa. We write A B and say A and B are equivalent or matching sets. Example. Definition. (1) Set A is a subset of set B, written A B, if and only if every element of A is also an element of B. Example. {1, 2, 4} {1, 2, 3, 4, 5}, but {1, 2, 3} 6 {1, 2, 4} Note. For every set A, A A and ; A (Why?) (2) Set A is a proper subset of set B, written A B, if A B and A 6= B (i.e., B has an element not in A). Example. {1, 2, 4} {1, 2, 3, 4, 5}, but {1, 2, 3} 6 {1, 2, 3}

3 2.1. SETS AS A BASIS FOR WHOLE NUMBERS 13 Note. A = B if and only if A B and B A. Definition. A set is finite if it is empty or can be put in 1 1 correspondence with a set of the form {1, 2, 3,..., n} where n is a counting number. A set is infinite if it is not finite. Note. A set is infinite if and only if it is equivalent to a proper subset of itself. Example. {1, 2, 3, 4,... } l l l l {2, 4, 6, 8,... } What are some implications here? Is every infinite set equivalent to the set of counting numbers? Venn diagrams are used to show relationships between sets. We use a rectangle called the universal set U to represent all elements considered in a discussion or context and circles for other sets. A B This is a proper subset since x /2 A.

4 14 2. SET, WHOLE NUMBERS, AND NUMERATION Two sets are disjoint if they have no elements in common. Operations on sets: (1) The union of two sets A and B, denoted A [ B, is the set containing all elements belonging to A or to B (or to both). A [ B (2) The intersection of two sets A and B, denoted A \ B, is the set of all elements common to both A and B. A \ B

5 2.1. SETS AS A BASIS FOR WHOLE NUMBERS 15 (3) The complement of a set A, denoted A, is the set of all elements in the universe U not in A. A (4) The set di erence (or relative complement) of set B from set A, denoted A B, is the set of all elements of A that are not in B. A B (5) The Cartesian product of set A with set B, denoted A B and read A cross B, is the set of all ordered pairs (a, b) where a 2 A and b 2 B. Note. A [ B = {x x 2 A or x 2 B} A \ B = {x x 2 A and x 2 B} A = {x x 2 U and x /2 A} A B = {x x 2 A and x /2 B} A B = {(a, b) x 2 A and x 2 B}

6 16 2. SET, WHOLE NUMBERS, AND NUMERATION Problem (Page 56 # 25). Let A = {50, 55, 60, 65, 70, 75, 80}, B = {50, 60, 70, 80}, C = {60, 70, 80}, D = {55, 65}. (a) A [ (B \ C) = A [ {60, 70, 80} = A (b) (A [ B) \ C = A \ C = C (c)(a \ C) [ (C \ D) = C [ ; = C (d)(a \ C) \ (C [ D) = C [ {55, 60, 65, 70, 80} = C (e)(b C) \ A = {50} {50} \ A = {50} (f)(a D) \ (B C) = {50, 60, 70, 75, 80} \ {50} = {50}

7 2.1. SETS AS A BASIS FOR WHOLE NUMBERS 17 Problem (Page 56 # 39). A university professor asked his class of 42 students when they had studied for his class the previous weekend. Their responses were as follows: 9 had studied on Friday 18 had studied on Saturday 30 had studied on Sunday 3 had studied on both Friday and Saturday 10 had studied on both Saturday and Sunday 6 had studied on both Friday and Sunday 2 had studied on Friday, Satuday, and Sunday Assuming that all 42 students responded and answered honestly, answer the following questions. (a) How many students studied on Sunday but not on either Friday or Saturday? (b) How many students did all of their studying one one day? (c) How many of the students did not study at all for this class last weekend? Strategy 7 Draw a diagram. Here we use a Venn diagram (A = Friday, B = Saturday, C = Sunday). 2 had studied on Friday, Satuday, and Sunday 2

8 18 2. SET, WHOLE NUMBERS, AND NUMERATION 3 had studied on both Friday and Saturday 10 had studied on both Saturday and Sunday 6 had studied on both Friday and Sunday had studied on Friday 18 had studied on Saturday 30 had studied on Sunday (a) How many students studied on Sunday but not on either Friday or Saturday? 16 (b) How many students did all of their studying one one day? 25 (c) How many of the students did not study at all for this class last weekend? 2

9 2.2. WHOLE NUMBERS AND NUMERATION Whole numbers and numeration A number is an idea or abstraction that represents a quantity. A numeral is the symbol used to represent a number. There are three common uses of numbers: (1) A cardinal number describes how many elements are in a finite set. is the set of whole numbers. W = {0, 1, 2, 3,... } (2) The ordinal numbers deal with ordering: first, second, third, etc. There are three ways of ordering whole numbers: (a) the usual counting chant: 3 < 6 since 3 comes before 6 in the chant. (b) using 1-1 correspondence: We let n(a) represent the number of elements in a set A and n(b) represent the number of elements in a set B. Let a = n(a) and b = n(b). Then a < b or b > a if A is equivalent to a proper subset of B. Example.

10 20 2. SET, WHOLE NUMBERS, AND NUMERATION (c) using the whole number line. 4 < 8 since 4 is to the left of 8 on the whole number line. (3) Identification numbers are used to name things, such as telephone numbers, social security numbers, and CBU 899 numbers. Numeration systems use various numerals to represent numbers. (1) The tally numeration system uses single strokes for numbers and can be improved by grouping. For 37: or We see that grouping helps considerably. What are some advantages and disadvantages of this system? (2) The Egyption numeration system uses the following numerals: This is an additive system since the values of the individual numerals are added together to form numbers. Note. The order in which the numerals are written for a number is immaterial.

11 2.2. WHOLE NUMBERS AND NUMERATION 21 (3) The Roman numeration system uses the following basic numerals: It is a positional system in that the position of a numeral a ects the value being represented. For example, IX is di erent from XI. It is also a subtractive system in that a lower numeral to the left of a numeral means subtraction rather than addition: IV means 5-1 or 4. Here are some common pairings: Finally, this is also a multiplicative system in that XV means or 15,000. Example. VMCMXLXIV means V M CM XL IV = = 6944.

12 22 2. SET, WHOLE NUMBERS, AND NUMERATION (4) The Babylonian numeration system uses the following two symbols: This system uses place value, where symbols represent di erent values depending on the place in which they were written. Place value is based on 60. But consider the following: Does this represent = 74 or (60) + 14 = 3614? To help in cases like this, the placeholder was introduced to indicate a vacant place. (What number seems to be missing in this and all previous systems?) With this new symbol, means = 74, while means (60) + 14 = 3614.

13 2.2. WHOLE NUMBERS AND NUMERATION 23 (5) The Mayan numeration system was a vertical place value system and had a symbol for zero. The three symbols used are Here are some sample simple numbers: This last number is di erent in that it occupies two levels, the zero for ones and the dot for one twenty. Remember, this is a vertical system. This system also had a varying base: (1 7200) + (6 360) + (0 20) + (3 1) = = 9363 Express the following in our numeration system:

14 24 2. SET, WHOLE NUMBERS, AND NUMERATION The answers in our numeration system: The following table summarizes the attributes of the numeration systems we have studied: 2.3. The Hindu-Arabic System This is our system. Features: (1) 10 symbols can be used to represent all whole numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. (1A) Suppose we only used 5 symbols 0, 1, 2, 3, 4. (2) Grouping by 10 s ours is a base 10 system. (a) Bundles of sticks model each group of 10 sticks is bundled with a rubber band.

15 2.3. THE HINDU-ARABIC SYSTEM 25 Ten bundles of ten are bundled to make 100, etc. (b) Base ten pieces: blocks, 4 flats, 2 longs, 7 units = 3(1000) + 4(100) + 2(10) + 7(1). (2A) Grouping by fives base 5. Write this as 32five or 325 and read as three two base 5. Some block representations of base five numbers follow.

16 26 2. SET, WHOLE NUMBERS, AND NUMERATION The first 10 base five numerals follow: Adding 15 to a numeral follows in the block model. (3) We have place value, which implies the system is positional. (3A) The same for base five. (4) The system is additive and multiplicative. A numerals s expanded form or expanded notation expresses a numeral as the sum of its digits times their respective place values. 75, 234 = 7(10, 000) + 5(1000) + 2(100) + 3(10) + 4(1) (4A) The same for base 5. 23, 2345 = 2(10, 0005) + 3(10005) + 2(1005) + 3(105) + 4(15) = 2(625) + 3(125) + 2(25) + 3(5) + 4(1) = This serves as an example of converting base five to base ten.

17 2.3. THE HINDU-ARABIC SYSTEM 27 Example (Converting from base ten to base five). Convert 7326 to base five. 5 0 = 1, 5 1 = 5, 5 2 = 25, 5 3 = 125, 5 4 = 625, 5 5 = 3125, 5 6 = 15625,... Example. Convert 7326 to base = = 1, 7 1 = 7, 7 2 = 49, 7 3 = 343, 7 4 = 2401, 7 5 = 16807, = Each Hindu-Arabic numeral has an associated name. Note. The word and does not appear in any name.