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What is a set? Sets and WholeNumber Operations and Properties Section 2.1 Sets and Whole Numbers Types of Sets A set with no elements is called the _________, or null set, and is denoted by { } or Ø. n( { } ) = n(Ø) = 0 Example: The set of all whales that are not mammals. A set with a limited number is called a _______ _________. Example: The set of people in this classroom. A set with an _________ number is called an infinite set. Example: The set of whole numbers. Sets A and B are _________ _________, A=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. (Must have exactly the same elements.) n(A) = n(B) Example: T = {t,e,a,c,h,e,r} and C = {c,h,e,a,t,e,r} T=C A set is any _________ of objects or ideas that can be listed or described. Example: The set of whole numbers W = { 0, 1, 2, 3, 4, 5, …} Each individual object in a set is called an _________ of the set. One-to-One Correspondence Equal Sets Sets A and B have a __________________ correspondence if and only if each element of A can be paired with exactly one element of B and each element of B can be paired with exactly one element of A. Example: Individuals and Social Security Numbers Equivalent Sets Sets A and B are _________ _________, A~B, if and only if there is a one-to-one correspondence between A and B. same or different types of elements n(A) = n(B) Example: U = {red, white, blue} and S = {1, 2, 3} U~S 1 Whole Numbers and Sets Subsets A _________ _________is the unique characteristic embodied in each finite set and all the sets equivalent to it. The number of elements in set A is expressed as n(A). When _________ two whole numbers,, you y can look at sets for each of the numbers. If a one-toone correspondence cannot be made between the elements of two sets, the set with elements left over is said to have more elements than the other set and the whole number for that set is greater than that of the other set. Greater than > Less than < Two Types of Subsets For all sets A and B, A is a _________ of B, symbolized as A ⊆ B, if and only if each element of A is also an element of B. Example: U = {square, circle, rectangle, triangle} A = {circle, triangle, rectangle} proper subset B = {triangle, square, rectangle, circle} improper subset Determining the Number of Subsets {1} {1,2} c) {1,2,3} d) {1,2,3,4} Write a rule for the number of subsets with n elements. a) A _________ subset identifies a subset that contains part, but not all, of the elements of a set. If X is a subset of Z,, then Z contains more elements than X. An _________ subset is a subset that contains all the elements of the set. If X is a subset of Z, then X is equal to Z. b) Write a rule for the number of proper subsets with n elements. Number Sets (continued) Number Sets Whole Numbers: the individual whole numbers, including _________, that comprise a single set of infinite numbers. W = { 0, 1, 2, 3,…} Natural (Counting) Numbers: the infinite set of whole numbers, excluding zero, that is used in _________. N = {1, 2, 3, 4, …} Integers: the infinite set of _________ whole numbers, negative numbers, and zero. I = {…, -3, -2, -1, 0, 1, 2, 3, …} List all subsets of: _________ Numbers: the infinite set of positive and negative numbers that can be described as a comparison of two integers. (Fractions, repeating decimals, terminating decimals) Q = {…, -1, -¾, -.15, 0, ¼, ½, ⅞, 1, …} _________ Numbers: the infinite set of positive and negative numbers that cannot be expressed as a comparison between two numbers. (non-repeating, non-terminating) _________ Numbers: the infinite set of numbers that include the rational numbers and the irrational numbers. 2 Real Numbers Irrational Numbers Rational Numbers Integers Whole Numbers Natural (Counting) Numbers Classroom Activity ROLEPLAY! Using a set of nesting boxes, have student volunteers come to the front of the room and illustrate the REAL NUMBER SYSTEM. Activity Materials 6 boxes: 4 that will stack inside each other, a 5th that when put with 4 nested will all fit inside largest, g , 6th box Label each box with the appropriate name. (Real Numbers, Irrational Numbers, Rational Numbers, Integers, Whole Numbers, Natural Numbers) 3