Sets and Venn Diagrams By Amber K. Wozniak

What is a Venn Diagram? A Venn diagram is a way to show the relationship between two or more sets. Everything inside the black outline of the rectangle is defined as the “universal set” or U. Set U is the main set and then there are subsets. The blue circle is one subset and the purple is another subset of U.

The Diagram (continued) To clarify: Each circle is a set representing its own collection of elements. The purpose of the Venn diagram is to show the relationship between any two or more sets, usually using the overlapping of the circles.

How are Sets Related? Sets are said to be in union when they are overlapping, like at the right. Saying “union” includes all elements in A, B, or in both A and B. To write A union B, it would be AUB The section between the two sets, where elements are in A and B, is known as the intersection. Set A Set B Intersection Elements in A and B are represented by writing “A B= {…}” with the elements inside the braces.

Sets can also be disjoint sets Sets can also be disjoint sets. That is, they can have no common elements. On the other hand, sets can be equal if they contain all of they same elements. One set can not have any elements the other does not or they are not equal sets.

Example 1 Let’s say set A is all odd numbers 1 to 10 This means A= {1, 3, 5, 7, 9} Let’s say there is also a set B that is all multiples of 3 from 3 to 10 This mean B= {3, 6, 9} Set B Set A 7 6 9 1 5 3 U

To define the Venn diagram on the previous slide we would write: U is the universal set and includes ALL elements within the universe, or the outline of the rectangle. A= {1, 3, 5, 7, 9} B= {3, 6, 9} A B= {3, 9} This tells us that the elements 3 and 9 are in both sets A’= {6} This tells us all the elements NOT in A The ‘ means the “compliment” of A, or all the elements that you would need to add to set A to give you everything in the universal set U (everything inside the black outline). B’= {1, 5, 7} This tells us all the elements NOT in B

Example 2 Set T GIVEN: Set U is all whole numbers to 10, set R is all multiples of 5 from 5 to 10, and Set T is all multiples of 2 from 2 to 10. Set R U U= {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, R= {5, 10}, and T= {2, 4, 6, 8, 10} Look for the elements that R and T have in common; those are the elements that will go in the intersection, or R T.

They only element that R and T have in common is 10, so: R T= {10} Set T 10 Set R That leaves 5 in the crescent of R. To write this it is: T’ U R= {5} (this tells is that 5 is NOT and element of T, but it is an element of R) U U= {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, R= {5, 10}, and T= {2, 4, 6, 8, 10} You can use the same reasoning to determine that 2, 4, 6, and 8 are all elements in the crescent part of T. R’ U T= {2, 4, 6, 8} (this tells us, again, that these four elements are NOT part of R, but they are included in T)

Set T 6 You must notice that not all of the elements are accounted for in the universe. What elements are left and where do they go? 2 5 10 8 4 Set R U U= {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, R= {5, 10}, and T= {2, 4, 6, 8, 10} U includes all whole numbers through 10 and there are only six elements in the Venn diagram above. The numbers 0, 1, 3, 7, and 9 are missing yet. Since these numbers do not belong in either R or T, they must go outside R U T. To write that it is the “compliment of (R U T)” (everything outside of both R and T). (R U T)’={0, 1, 3, 7, 9}

0 1 Set T 7 6 After all that, this would be your completed Venn Diagram with all the elements in their correct set(s). 2 5 10 8 4 Set R 3 9 U To tell that one number is a included in a set, you write is using an epsilon (E). To write that 5 is included in the set R, you would write: 5 E R To write that 5 is NOT included in the set T, you would write: 5 E T So, you could say 5 E R, but 5 E T

It’s Quiz Time Make sure that you feel comfortable with the Venn diagram. If you are unsure, look back at the information provided here OR Visit some of the web sites available on the next slide. Take the Sets and Venn Diagrams Quiz Answers to the Quiz

Links to Useful Web Sites Venn Diagrams (Shodor.org) Venn Diagrams A Survey of Venn Diagrams JOMA.org