COLLEGE ALGEBRA. Intro, Sets of Real Numbers, & Set Theory

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1 COLLEGE LGER y: Sister Mary Rebekah Cornell-Style Fill in the lank Notes and Teacher s Key Intro, Sets of Real Numbers, & Set Theory 1

2 Vocabulary Workshop SIMPLIFY Expressions EVLTE Expressions SOLVE Equations EVLTE Functions IDENTIFY 2

3 Vocabulary Workshop SM & ddend DIFFERENCE PRODCT & factor QOTIENT DIVIDEND & divisor 3

4 Real Numbers and Sets of Numbers The Set of REL NMERS The Real Numbers consist of the following sets of numbers: Irrational Numbers: Non-terminating, non-repeating decimals. ie, Rational Numbers: Terminating or repeating decimals. ll rational numbers can be written as fractions. o Integers o Whole Numbers o Natural Numbers Organizing REL NMERS SET LETTER EXMPLES Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers SKILLS Practice Directions: Name all the sets to which the following values belong π π π 4

5 Set Notation Element Notation If x is a member or element of the set S, we write If x is a not an element of the set S, we write Set Notation Type of Notation Description Example Set Notation Interval Notation Set-uilder Notation Verbal Notation SKILLS Practice Directions: Fill in the blanks with the correct notation for the given sets. Set Notation Interval Notation Set-uilder Verbal Notation 1. { -5,0,5, } 2. S={x x = 2n,for n Z} 3. (, 4) 4. ll integers between -10 and 10, exclusive 5

6 Set Notation Practice with Set Notation Directions: Determine which are equivalent sets in the following groups of sets. 1. Which of the following sets are equal to the set of all integers that are multiples of 5? There may be more than one or none. 1) {5n n R} 2) {5n n Z} 3) {n Z n = 5k and k Z} 4) {n Z n = 5k and n Z} 5) { 5,0,5,10} 2. Which of the following sets are equal to the set of all negative real numbers? There may be more than one or none. 1) {n n R, n < 0} 2) {n n Z, n < 0} 3),0 4) {, -3, -2, -1} 2. Which of the following sets are equal to the set of all positive integers? There may be more than one or none. 1) {n n N} 2) {n n Z, n > 0} 3) 0, 4) {1, 2, 3, 4, } 5) {n n Z} Null Set Subset Examples Given the set S={x x = 2n,for n Z}, a) Write this set using ellipses b) Write this set using regular words c) Write two subsets of this set. 6

7 Set Operations Definition of nion The union of sets and, denoted by (read: union ), is the set consisting of all elements that belong to. In symbols Examples of nion Example 1. Venn Diagram. Color. Example 2. If = {1,2,3,4,5} and = { 4,5,6,7,8,9,10}, please write. Definition of Intersection Def. of intersection. The intersection of sets and, denoted by (read: intersection ), is the set consisting of all elements that belong to. In symbols Examples of intersection Example 3. Venn Diagram. Color. Example 4. If = {1,2,3,4,5} and = { 4,5,6,7,8,9,10}, please write. 7

8 Set Operations Disjoint Sets Disjoint sets have no elements in common. The intersection of disjoint sets is the empty set, or null set. The diagram below shows two disjoint sets. Examples of Disjoint Sets Example 1. If = {1,2,3,4} and = { 5,6,7,8,9,10}, please write. Example 2. What is the intersection of all rational numbers and all irrational numbers? Example 3. What is the intersection of all even numbers and all odd numbers? Example 4. What is the intersection of all negative integers and all positive integers? Practice Set C = x x is a natural number less than 20 Set D = y y is an odd integer Set E = z z is a multiple of 4 1. Write C D. 2. Write C E. 3. Write D E. 4. Write C in set builder notation. 5. Write D in set builder notation. 6. Write E in set builder notation. 8

9 Set Operations Set Complements set complement, written, refers to anything NOT in set. Practice with Set complements VENN DIGRMS ( ) ( ) Practice Let = 2,5 Let = 5,7,9 Let C = x x is an odd whole number less than 9 Let D = x x is an even whole number less than 9 Let =the set of whole numbers less than 9 Directions: Find each union or intersection using the defined sets,, C, and D C 3. D 4. C 5. D 6. D 7. C 8. D 9. 9

Set Operations Practice Directions: Draw a Venn Diagram to represent the union and intersection of the given sets. 1. lex has cats, rabbits, and fish as pets. ecky has cats and dogs.

10 Set Operations Practice Directions: Draw a Venn Diagram to represent the union and intersection of the given sets. 1. lex has cats, rabbits, and fish as pets. ecky has cats and dogs. Corry has cats, birds, fish and turtles. Let = {cats, rabbits, fish}, = {cats, dogs}, C={cats, birds, fish, and turtles} 2. Let X = x x is a letter in the word LGER Y = y y is a letter in the word GEOMETRY Z = z z is a letter in the word CLCLS pplication Directions: Respond to the following situations involving sets. 3. Find two sets and such 4. Let M = x 3x, x Z and that = 1,2,3,4,5 and N = x 4x, x Z, describe the = 2. intersection between M and N. 5. Set X has 10 elements, set Y has 15 elements, and X Y has 5 elements. How many elements are in X Y? 6. Find two sets and such that = 1,2,3,4,5 and =. True or False Directions: Identify each statement as true or false. se a Venn diagram or give a counter example to justify your answer. 1. The intersection of two sets is always a subset of their union. 2. Two sets that contain no elements in common are disjoint sets. 3. The intersection of the set of even numbers and the set of prime numbers is the empty set. 10

Introduction. Sets and the Real Number System

Introduction. Sets and the Real Number System Sets: Basic Terms and Operations Introduction Sets and the Real Number System Definition (Set) A set is a well-defined collection of objects. The objects which form a set are called its members or Elements.