Solving Problems using Venn Diagram Mr. Albert F. Perez June 29, 2015

Venn Diagram and Set Notation The following examples should help you understand the notation, terminology, and concepts related to Venn diagrams and set notation. Let U = {1, 2, 3, 4} A = {1, 2} and B = {2, 3} Then we have the following relationships, with pinkish shading marking the solution "regions" in the Venn diagrams:

Set Notation PronunciationMeaningVenn diagramAnswer A U BA U B "A union B" everything that is in either of the sets {1, 2, 3} "A intersect B" only the things that are in both of the sets {2} A c or A’ "A complement", or "not A" everything in the universe outside of A {3, 4} Venn Diagram and Set Notation Let U = {1, 2, 3, 4} A = {1, 2} and B = {2, 3}

Venn Diagram and Set Notation "A minus B", Or "A complement B" everything in A except for anything in its overlap with B {1} (A U B)’ " not (A union B)" everything outside A and B {4} " not (A intersect B)" everything outside of the overlap of A and B {1, 3, 4} Set Notation PronunciationMeaningVenn diagramAnswer ( A  B)’ Let U = {1, 2, 3, 4} A = {1, 2} and B = {2, 3} A’  B

Venn Diagram and Set Notation Describe the following sets: Wet clothes and dry clothes are separated A B Set A and B are disjoint sets All cheerleaders are girls Set D is a subset of Set C

Venn Diagram and Set Notation Try this: Describe the following sets: Some boys and some girls are varsity players The intersection of Set E and Set F is composed of Varsity players

Venn Diagram and Set Notation Observe the following Venn diagram and describe by providing the appropriate set operation.

Venn Diagram and Set Notation Observe the following Venn diagram and describe by providing the appropriate set operation.

Venn Diagram Word Problems Venn diagram word problems generally give you two or three classifications and a bunch of numbers. You then have to use the given information to populate the diagram and figure out the remaining information. For instance… Out of forty students, 14 are taking English Composition and 29 are taking Chemistry. If five students are in both classes: How many students are taking English Composition only? How many students are taking Chemistry only? How many students are in neither class? How many are in either class?

Venn Diagram Word Problems First, draw a universe for the forty students, with two overlapping circles labeled with the total in each: Out of forty students, 14 are taking English Composition and 29 are taking Chemistry. If five students are in both classes: How many students are taking English Composition only? How many students are taking Chemistry only? How many students are in neither class? How many are in either class?

Venn Diagram Word Problems Since five students are taking both classes, put "5" in the overlap: Out of forty students, 14 are taking English Composition and 29 are taking Chemistry. If five students are in both classes: How many students are taking English Composition only? How many students are taking Chemistry only? How many students are in neither class? How many are in either class?

Venn Diagram Word Problems We have now accounted for five of the 14 English students, leaving nine students taking English but not Chemistry, so we put "9" in the "English only" part of the "English" circle: Out of forty students, 14 are taking English Composition and 29 are taking Chemistry. If five students are in both classes: How many students are taking English Composition only? How many students are taking Chemistry only? How many students are in neither class? How many are in either class?

Venn Diagram Word Problems We have also accounted for five of the 29 Chemistry students, leaving 24 students taking Chemistry but not English, so we put "24" in the "Chemistry only" part of the "Chemistry" circle: Out of forty students, 14 are taking English Composition and 29 are taking Chemistry. If five students are in both classes: How many students are taking English Composition only? How many students are taking Chemistry only? How many students are in neither class? How many are in either class?

Venn Diagram Word Problems This tells us that a total of = 38 students are in either English or Chemistry (or both). This leaves two students unaccounted for, so they must be the ones taking neither class. Out of forty students, 14 are taking English Composition and 29 are taking Chemistry. If five students are in both classes: How many students are taking English Composition only? How many students are taking Chemistry only? How many students are in neither class? How many are in either class?

The Venn diagram is also used to analyse different relationship that exists between two or more objects. 1.Into swimming but not basketball? 2. Into both swimming and basketball? 3. Into either swimming or basketball? 4. Neither into swimming nor basketball? Venn Diagram Word Problems The Venn diagram below shows student involvement in two sports. If 100 students were surveyed, how many students were:

140 students were surveyed. The Venn diagram shows the number of students who enjoy singing and/or dancing. How many students enjoy: 1.Dancing but not singing? 2. Singing or dancing? 3. Singing and dancing? 4. Neither singing nor dancing? Venn Diagram Word Problems

Fifty people are asked about the pets they keep at home. The Venn diagram shows the result. Let D = { people who have dogs} F = { people who have fish } C = { people who have cats } Venn Diagram Word Problems How many people have a. Dogs? b. Dogs and fish? c. Dogs or cats? d. Fish and cats but not dogs? e. Dogs or fish but not cats? f. All three? g. Neither one of the three?

Venn Diagram Word Problems 100 people were surveyed. Suppose: B = { people who like beaches } M = { people who like mountains } C = { people who like big cities } How many people like staying in: 1. big cities only? 2. mountains or beaches? 3. mountains and beaches but not big cities? 4. cities or beaches but not mountains? 5. all three? 6. Neither of the three?

In a high school, students know at least one of the dialects, Tagalog and Cebuano. 201 students know Tagalog, 114 students know Cebuano and 100 of them know both Tagalog and Cebuano. How many students are there in the school? Venn Diagram Word Problems

Solution CebuanoTagalog

Problem # 2 In a class of 15 boys, there are 10 who play basketball and 8 play chess. a.How many play both of the games? b.How many play basketball only? c.How many play chess only?

Solution 108 Total Numbers of boys = 15

Solution 735

Problem # 3 In a class, 40 can speak either English or Tagalog or both. If 25 students can speak English and 20 can speak both; find the number of those who can speak Tagalog only. 15 students

Solution

Historical Note John Venn (4 August 1834 – 4 April 1923), was a British logician and philosopher. He is famous for introducing the Venn diagram, which is used in many fields, including set theory, probability, logic, statistics, and computer science.