Topic 7: Venn Diagrams, Sets, and Set Notation

Transcription

1 Topic 7: Venn Diagrams, Sets, and Set Notation for use after What Do You Expect? Investigation 1 set is a collection of objects. These objects might be physical things, like desks in a classroom, or ideas, like types of quadrilaterals or positive even numbers. Each object is called an element of the set or a member of the set. For example, the number 1 is an element of (or belongs to) the set of positive even numbers. The symbol means is an element of, and the symbol means is not an element of. 1 the set of positive even numbers 17 the set of positive even numbers Sets can be specified by describing the elements or by listing the elements inside braces. Set of positive even numbers = {,,,, 10,...} In this case the list continues indefinitely, since the list of positive even numbers goes on forever. If there is a last element in a set, that element can be listed after the ellipsis. Set of positive even numbers less than 100 = {,,,,...} When a set has only a small number of elements, it is usually easiest simply to list all of the elements. Set of one-digit positive numbers divisible by = {0,,,, } subset of a set is a set that contains some or all of the elements of the set, but no other elements. The symbol is used to indicate is a subset of. The symbol is used to indicate is not a subset of. {0,, } {0,,,, } {0,,, 5} {0,,,, }

2 You learned about Venn diagrams in Prime Time. Venn diagrams can be used to show relationships among sets. The union of two sets is a set that contains all of the elements of both sets. ny elements that belong to both sets are listed only once. The symbol is used to designate the union of two sets. Shading is used to show everything that should be included in the union. {0,, } {,, } = {0,,,, } 0 {1, 3, 5, 7, } {0,,,, } = {0, 1,, 3,, 5,, 7,, } Problem List the different subsets of {0, 1,, 3, } with one element.. Is {0, 1, } the same as {, 1, 0}? Explain. 3. List the different subsets of {0, 1,, 3, } with four elements.. How are your answers for parts (1) and (3) related?. 1. What is {0, 1, } {, 3, }? Explain.. What is {0, 1, } {3, }? 3. What do you notice about the two answers? Explain.

3 . Use braces to describe each set. 1. Positive Odd Numbers. Whole Numbers 3. Integers D. 1. Draw a Venn diagram to show how the Positive Odd Numbers and Whole Numbers are related.. Draw a Venn diagram to show how the Positive Odd Numbers and Integers are related. set with no elements is called the Empty set = {} empty set. The intersection of two sets is a set that contains only those elements that are in both sets. The symbol is used to indicate an intersection. {0,, } {,, } = {} {,, } {0,,,, } = {,, } Venn diagrams can be used to show the intersection of two sets. {0,, } {,, } 0 Sometimes the intersection of two sets is the empty set. {0,,,, } {1, 3, 5, 7, } = {} On a Venn diagram, this looks like the diagram below

4 The complement of a set is all of the elements not in the set. The complement of set is shown as and is read, prime. The complement of a set can only be determined relative to the set of all things being considered. When considering all whole numbers, if is the set of all even numbers, is the set of all odd numbers. Problem What is the intersection of the set of all multiples of and the set of all multiples of 3?. Define two sets whose intersection is {5, 10, 15, 0}. 3. What is the intersection of the set of regular polygons and the set of quadrilaterals? Explain.. For parts (1) (3), use sets P and Q. P = {,,,, 10} Q = {3,, } 1. Use set notation to show the union of P and Q.. Use set notation to show the intersection of P and Q. 3. Draw a Venn diagram for each of your sets in parts (1) and ().. For parts (1) (), use sets P and Q from Question. onsider only whole numbers from 1 to P Q. P Q 3. (P Q). P Q D. Describe each set. 1. complement of rational numbers considering the set of real numbers. whole numbers rational numbers 3. rational numbers irrational numbers

5 Exercises Use set notation to write the union and the intersection for each pair of sets. 1. {, 3,, 5,, 7} {, 7,, }. {0,,,,,...} {1,3,5,7,,11,...} U... V U... V,, 3, 1 3, 1, 1 5, 1, 10,. {1,,,, 1, 3,...} {1,,1,,5,...} 5. Draw a Venn diagram to show each of your answers to Exercise 1. onsider the integers from 1 to Describe the complement of {3,,, 1, 15, 1,... } 7. Describe the complement of {,,,, 10,...100} For Exercises 11, copy the Venn diagram below and shade the region indicated.. intersection of,, and. union of and 10. intersection of with the union of and 11. union of with the intersection of and Decide whether each statement is true or false. Explain. 1. The intersection of X and Y is a subset of X. 13. X is a subset of the union of X and Y. 1. X is a subset of the intersection of X and Y. 15. {0,,,,,...} 1. {0,,,,,...}

6 Topic 7: Venn Diagrams, Sets, and Set Notation t a Glance PING 1 day Mathematical Goals Demonstrate relationships among sets through the use of Venn diagrams Determine subsets, complements, intersection, and union of sets Use set notation to denote elements of a set Guided Instruction Topic 7 is an introduction to informal set theory. The terminology and notation used here is likely to be new information for many students. It is important that the instruction does not become a lesson only on terminology and notation. Use of Venn diagrams will help many students internalize the ideas and provide a way for continued development of visualization skills. Some students may find working with finite sets easier than working with infinite sets, such as the set of integers. The elements of a small finite set can all be listed, so there is no ambiguity about what those elements are. The elements of an infinite set cannot all be listed; students must consider elements that are not actually written down. This is more difficult, so provide ample time for students to work and for discussion of answers. Understanding the empty set is frequently a challenge for students. The empty set has no elements; it is empty. This is different from the set containing the element 0; that is, {0}, which is not empty (since it contains one element). The difficulty is that some students think of zero as nothing. However, 0 is a number; it is an object; it is something. e patient and persistent in helping students understand that { } and {0} are not the same. fter the Problem 7.1 Introduction, ask: What other words might mean almost the same thing as union? fter defining empty set, ask: Is the empty set nothing? How many elements are in the empty set? True or false? The empty set is a subset of every set. Why? fter the discussion about intersection on page 1, ask: How is the intersection of two sets similar to the intersection of two streets? What do you know about two sets whose intersection is the empty set? What two sets have the set of multiples of 10 as their intersection? If X is the set of multiples of a number n and Y is the set of multiples of a number m, what is the intersection of X and Y? Why? Vocabulary set element subset Venn diagrams union empty set intersection complement ssignment Guide for Topic 7 ore 1 1

7 nswers to Topic 7 Problem {0}, {1}, {}, {3}, {}. Yes; each set contains the same elements. 3. {0, 1,, 3}, [0, 1,, }, {0, 1, 3, }, {0,, 3, }, {1,, 3, }. There are the same number of four-element subsets in part (3) as there were oneelement subsets in part (1).. 1. {0, 1,, 3, }; the union of two sets contains all of the elements from both sets, but common elements are listed only once.. {0, 1,, 3, } 3. The answers are the same; original sets in parts (1) and () contain all the same elements.. 1. {1, 3, 5, 7,, 11, 13,...}. {0, 1,, 3,, 5,, 7,,,...} 3. {...,, 1,0,1,,...} D. 1. Whole Numbers. Problem 7. Positive Odd Numbers Integers Positive Odd Numbers. Possible answers: {5, 10, 15, 0, 5} f {0, 5, 10, 15, 0}. The two sets must each contain 5, 10, 15, and 0 as elements, with no other common elements. 3. {square}; a square is the only regular quadrilateral.. 1. P e Q = {, 3,,,,, 10}. P f Q = {} 3. P Q 3 10 P. 1. P f Q = {3, }. P f Q = {,,, 10} 3. (P f Q) = {1,, 3,, 5, 7,,, 10}. P e Q = {1,, 3,, 5, 7,,, 10} D. 1. Irrational numbers; that is, every number that is NOT a rational number.. Whole numbers, since whole numbers are a subset of the rational numbers. 3. The empty set, since there is NO number that is both rational and irrational. Exercises union: {, 3,, 5,, 7,, } intersection: {, 7}. union: {0, 1,, 3,, 5,, 7,...} intersection: { } union: {,,,,,,,,,,,,, } intersection: {,, }. union: {1,,,, 1, 3,, 1,...} intersection: {1,, 1,, 5,...} 3 Q. 1. {0,, 1,,,...}; multiples of = {0,,,,,...} and multiples of 3 = {0, 3,,, 1,...}.

8 The set of integers from 1 to 100 NOT divisible by 3 7. The set of odd integers from 1 to True; every element of the intersection must be an element of X. 13. True; the union contains every element of each set, and in particular every element of X. 1. Generally false; some elements of X might not also be elements of Y. ut, there are examples of X and Y for which X IS a subset of the intersection of X and Y. 15. True, since is an even number. 1. False, since is not an even number.. 10.