Chapter 2: Sets and Whole-Number Operations and Properties

2.1 Sets and Whole Numbers

2.1.1. Sets and Their Elements

2.1.1.1. Sets and their elements

2.1.1.1.1. set: any collection of objects or ideas that can be listed or described

2.1.1.1.1.1.1. Examples – primary colors, secondary colors, etc.

2.1.1.1.1.2. objects listed in a set are within braces: { }

2.1.1.1.1.2.1. Examples – {red, blue, yellow} or {orange, green, purple}

2.1.1.1.2. element: each individual object in a set, they are separated by commas

2.1.1.1.2.1.1. Examples – red is an element of the primary colors set; green is an element of the secondary colors set

2.1.1.2. One-to-one Correspondence

2.1.1.2.1. Definition of One-to-one Correspondence: Sets A and B have one-to-one correspondence if and only if each element of A can be paired with the exactly one element of B and each element of B can be paired with exactly one element of A.

2.1.1.2.1.1. See examples p. 52

2.1.1.2.2. Your turn p. 53: Do the practice and the reflect

2.1.1.3. Equal and Equivalent Sets

2.1.1.3.1. Definition of Equal Sets: Sets A and B are equal sets, symbolized by A = B (read “A is equal to B”), if and only if each element of A is also an element of B and each element of B is also an element of A.

2.1.1.3.1.1. Example – A = {a, b, c} and B = {b, c, a}, then A = B

2.1.1.3.2. Definition of Equivalent Sets: Sets A and B are equivalent sets, symbolized by A ~ B (read “A is equivalent to B”), if and only if there is one-to-one correspondence between A and B.

2.1.1.3.2.1. Example - A = {a, b, c} and B = {(, $, J}, then A ~ B

2.1.1.3.3. Note – All equal sets are equivalent sets, but only some equivalent sets are also equal sets

2.1.1.3.4. Your turn p. 54: Do the practice and the reflect

2.1.1.4. Subsets and Proper Subsets

2.1.1.4.1. Definition of a subset of a set: For all sets A and B, A is a subset of B, symbolized as A Í B, if and only if each element of A is also an element of B.

2.1.1.4.2. Venn diagram: representation of sets using circles, where the elements of a set are contained within a circle

2.1.1.4.3. Definition of a proper subset of a set: For all sets A and B, A is a proper subset of B, symbolized A Ì B, if and only if A is a subset of B and there is at least one element of B that is not an element of A.

2.1.1.4.4. A is not a subset of B is symbolized: A Ë B

2.1.1.4.5. See example 2.4 p. 67-68

2.1.1.5. The Universal Set, the Empty Set, and the Complement of a Set

2.1.1.5.1. The universal set, U, is made up of all of the possible elements that could be considered for a given situation. The universal set is either given or assumed from the context of the problem.

2.1.1.5.2. empty set or null set: a set with no elements in it can be designated by either of the following symbols, but NEVER both at the same time – { } or Æ but NEVER {Æ}

2.1.1.5.2.1.1. Examples – The set of 300 year old living cats; the set of unsuccessful students in this class J

2.1.1.5.3. Definition of the complement of a set: The complement of a set A, written A’ or , consists of all elements in U that are not in A.

2.1.1.5.4. See examples p. 56-57

2.1.1.5.5. Your turn p. 57: Do the practice and the reflect

2.1.2. Using Sets to Define Whole Numbers

2.1.2.1. finite set: a set with a limited countable number of elements

2.1.2.1.1.1. Examples – the amount of money in your pocket; the number of digits on your left hand; the number of red cells in your body, the set of digits, etc.

2.1.2.2. infinite set: a set with an unlimited number of elements

2.1.2.2.1.1. Examples – Number of points contained in a line; all of the even numbers; all of the odd numbers; etc.

2.1.2.3. Definition of a Whole Number: A whole number is the unique characteristic embodied in each finite set and all sets equivalent to it. The number of elements in set A (the cardinality of set A) is expressed as n(A).

2.1.2.3.1. Cardinality: the number of elements in a set: n(A) is some whole number

2.1.2.3.2. Example – A = {(, $, J} and B = {blue, gum, reading, playing}, then n(A) = 3 and n(B) = 4

2.1.2.4. The set of whole numbers is an infinite set designated in the following way: W = {0, 1, 2, 3, …}

2.1.2.5. Counting: the process that enables people systematically to associate a whole number with a set of objects

2.1.3. Using Sets to Compare and Order Whole Numbers

2.1.3.1. Procedure for using one-to-one correspondence to compare whole numbers

§ Look at the sets for each of the numbers

§ One-to-one correspondence cannot be made between the elements of two sets

· set with left over elements has more elements

· whole number for set with more is greater than for other set

2.1.3.1.1. See example 2.4 p. 60

2.1.3.1.2. Your turn p. 60: Do the practice and the reflect

2.1.3.2. Using Subsets to Describe Whole-Number Comparisons

2.1.3.2.1. Definition of less than and greater than: For whole umbers a and b and sets A and B, where n(A) = a and n(B) = b, a is less than b, symbolized by a < b, if and only if A is equivalent to a proper subset of B. Note that a is greater than b, written a > b, whenever b < a.

2.1.3.3. Ordering Whole Numbers

2.1.3.3.1. Increasing: the next whole number is 1 greater than the number it follows

2.1.3.3.2. n + 1

2.1.3.3.3. Decreasing: the next number is 1 less than the number that follows

2.1.3.3.4. n – 1

2.1.3.3.5. Trichotomy principle: For any two finite sets A and B, one of three things must be true –

· n(A) = n(B)

· n(A) > n(B)

· n(A) < n(B)

2.1.4. Important Subsets of Whole Numbers

2.1.4.1. Some special subsets of the set of whole numbers

2.1.4.1.1. Set of natural numbers

2.1.4.1.1.1. proper subset of the whole numbers

2.1.4.1.1.2. infinite set

2.1.4.1.1.3. Sometimes called the counting numbers

2.1.4.1.1.4. N = {1, 2, 3, …}

2.1.4.1.2. Set of even numbers

2.1.4.1.2.1. proper subset of the whole numbers

2.1.4.1.2.2. infinite set

2.1.4.1.2.3. E = {0, 2, 4, …}

2.1.4.1.3. Set of odd numbers

2.1.4.1.3.1. proper subset of the whole numbers

2.1.4.1.3.2. infinite set

2.1.4.1.3.3. O = {1, 3, 5, …}

2.1.4.1.4. There are just as many elements in W as there are in N, O, or E

2.1.4.2. Finding all the subsets of a finite set of whole numbers

2.1.4.2.1. See example 2.5 p. 62

2.1.4.2.2. Your turn p. 63: Do the practice and the reflect

2.1.4.2.3. Mini-investigation 2.4 – Finding a pattern

2.1.5. Three types of Numbers

2.1.5.1. Nominal: one, two, three, …

2.1.5.2. Ordinal: first, second, third, …

2.1.5.3. Cardinal: 1, 2, 3, …