Euler Diagram- Definition #ALG-2-18 Category Algebra Sub-category Set Theory Description An introduction to the Venn diagrams. Course(s) 085 091 093 095 097 100 ADN E18 PHT x x x x 1.3 x x x Mini-Lesson Definition(s) Sets A set is a collection of objects, which are called elements or members of the set. A set is well-defined if its contents can be clearly determined. In other words, the contents can be clearly named or listed based on fact and not opinion. Sets are generally named with capital letters. Euler diagram (pronounced “oy-ler”) As with a Venn diagram, in an Euler diagram, a rectangle usually represents the universal set, U. The items inside the rectangle may be divided into subsets of the universal set. Subsets are usually represented by circles. Euler diagrams best represent disjoint sets or subsets. Disjoint sets – Set A and set B have no elements in common. Subsets – When A ⊆ B, every element of A is also an element of set B. Euler diagram for number sets We often use Euler diagrams to represent the relationship between the common sets of numbers: Natural: 푁 = {1, 2, 3, 4, 5, … }, Whole: 푊 = {0, 1, 2, 3, 4, 5, … }, Integers: 푍 = {… , −3, −2, −1, 0, 1, 2, 3, … }, 푎 Rational: 푄 = { | 푎, 푏 ∈ 푍 푎푛푑 푏 ≠ 0}, Irrational Numbers , Real 푏 Numbers: 푅 = {푛푢푚푏푒푟푠 푡ℎ푎푡 푐표푟푟푒푠푝표푛푑 푡표 푝표푛푡푠 표푛 푛푢푚푏푒푟 푙푛푒} ALG-2-18 | Page 1 Rule Familiarize yourself with Euler Diagrams. Example ퟐ Consider the set of numbers: {−ퟏퟎ, −ퟎ. ퟓ, ퟎ, , 흅, ퟕ}. ퟑ Place each number in the Euler Diagram below. 7 흅 0 -10 ퟐ -0.5 ퟑ Remember! Euler diagrams are often used for disjoint sets or subsets. Practice Problems 1. Identify the set of numbers represented by each color in teh 2. Construct a Euler diagram for the following, placing each element 1 in the correct region: {−7, − , 0, √2, 3.1, 5} 2 3. Identify ALL of the a. Integers b. Rational numbers c. Irrational numbers d. Real numbers See also ALG-2-1: set - definitions ALG-2-7: common sets of numbers ALG-2-8: subset - definition ALG-2-16: visual representation of sets ALG-2-17: Venn Diagram ALG-2-18 | Page 2 Practice Answers 1. Red = Natural numbers Blue = Whole numbers Green = Integers Orange = Rational Numbers Yellow = Irrational Numbers Purple = Real Numbers 2. 5 √ퟐ 0 -7 ퟏ − 3.1 ퟐ 3. a. The integers include: 5, 0, −7 1 b. The rational numbers include: 5, 0, −7, 3.1, − 2 c. The irrational numbers include: √2 1 d. The real numbers include: 5, 0, −7, 3.1, − , √2 2 ALG-2-18 | Page 3 .