SET AND VENN DIAGRAMS

Transcription

1 SET AND VENN DIAGRAMS Part 1: Relationship among sets Sets: A set is a of the set. Examples of a set:. An individual object in a set is called a Venn diagrams: A Venn diagram is a to the relationships of sets. In a Venn diagram, circles are used to represent sets. There are 3 types of relationship between two sets A and B: 1. A is a of B, meaning that all members of A are also members of B. Examples with Venn diagrams: 2. A and B are, meaning that the two sets have no members in common. Examples with Venn diagrams: 1

2 2 1C SETS AND VENN DIAGRAMS 3. A and B are, meaning that the two sets have some common members. Examples with Venn diagrams: Exercise: Use the terms subset, disjoint and overlapping to describe the relationship of the following pairs of sets. Draw a Venn diagram to illustrate those relationships. (a) Mammals and repetiles; (b) Dancers and pianists; (c) Vehicles and things that have wheels; (d) Odd numbers and the set of integers greater than 5. Describe the relationships of numbers using Venn diagrams Describe the relationships of natural numbers, integers, rational numbers and real numbers with a Venn diagram: The set of natural numbers contains numbers. The set of integers contains numbers. The set of rational numbers contains numbers of the form. (e.g. ) The set of irrational numbers contains the number that cannot be expressed of the form. (e.g. ) The set of real numbers contains both and. The set of natural numbers is a of the set of integers. The set of integers is a of the set of rational numbers. The set of rational numbers is a of the set of real numbers. The Venn diagram looks like:

3 1C SETS AND VENN DIAGRAMS 3 Exercise Write down one member in each region of the Venn diagram above. Part 2: Categorical propositions A categorical proposition is a statement that claims the relationship of two sets. There are four standard forms of categorical propositions. 1. Example: Venn diagram: 2. Example: Venn diagram: 3. Example: Venn diagram: 4. Example: Venn diagram: Exercise Determine which type of standard form of the following categorical propositions is. Draw the corresponding Venn diagram. (a) All US presidents are men. (b) Some singers are not dancers. (c) No Democrat are Republicians.

4 4 1C SETS AND VENN DIAGRAMS Remark 1: The use of overlapping in describing a categoical proposition is slightly dierent from that in describing the relationship of two sets (Part 1). If two circles overlap, then those corresponding two sets might or might not share some members. In case 3, two sets share some members and we put a cross in the overlapping region to indicate they do so. However, in case 4, it is possible that those two sets do not share any members. Remark 2: Examples for Remark 2:

5 1C SETS AND VENN DIAGRAMS 5 Many categorical propositions are not in a standard form, but can be rewritten to a standard form. Examples Exercise Determine if the following sentence is a categorical proposition. If yes, put the categorical proposition in standard form with a Venn diagram. (a) Some Mathematics courses are interesting. (b) All people cannot walk. (c) Monkeys don't gamble. (d) How are you?

6 6 1C SETS AND VENN DIAGRAMS Part 3: Venn diagram with 3 sets Venn diagrams can also be used to visualize the relationships of two examples to illustrate how to describe the relationships with 3 sets. sets. Here we give Example Draw a Venn diagram with the three overlapping circles for the following three sets: adult, unemployed people and people who smoke. Describe the members of each region. Exercise Put a cross in the Venn diagram above to indicate the region that contains an unemployed teenage who do not smoke. Example Draw a Venn diagram with three overlapping cricles for the following three sets: published works, novels, and songs. Describe the members in each region or state that a regon has no members.

7 1C SETS AND VENN DIAGRAMS 7 Part 4: Venn diagram with numbers Example Given the following table of births with their birth weight status and mother's smoking status, draw a two-cricle Venn diagram that represents the results. low birth weight and smoking mother 18 normal birth weight and smoking mother 132 low birth weight and non-smoking mother 14 normal bith weight and non-smoking mother 186 Example All cyclists who competed in a race were given a drug test. Of the 18 who tested positive, 3 nished in the top 10. Twenty-ve cyclists tested negative. How many cyclists who tested negative did not nish in the top 10? How many cyclists were tested?

8 8 1C SETS AND VENN DIAGRAMS Example A survey revealed the following results about the news sourcs that a sample of 130 people use: TV only 20 TV and Internet only 12 Internet only? TV/radio and newspapers only 18 Newspapers only 15 Internet and newspapers only 22 None 6 All three sources 8 (a) Draw a three circle Venn diagram that summarized the results of the survey. (b) How many people use TV o newspapers? (c) How many people use TV or Internet? (d) How many people use TV or Inernet, but not newspapers? (e) How many people use Internet, but not TV? (f) How many people use TV, but not newspaper?