UNIT 1 MODULE 2
VENN DIAGRAMS AND SET OPERATIONS



A Venn diagram is a drawing that shows the general relationship between sets without listing any of the specific elements of the sets. Typically, in a Venn diagram we use figures such as circles and rectangles to represent the sets.



EXAMPLE 1.2.1

Let U = {Harpo, Groucho, Chico, Zeppo, Gummo, Karl, Skid, Hi, Lo}
S = {Harpo, Chico, Groucho}
V = {Harpo, Groucho}

Notice that V is a subset of S, and of course S and V are both subsets of U.

The Venn diagram below shows the correct relationship between the sets S, V and U.

Diagram showing circle V contained within circle S, contained within rectangle U





EXAMPLE 1.2.2

Let U = {Moe, Larry, Curly, Shemp, Curly Joe, Spurrier}
A = {Moe, Spurrier}
B = {Shemp, Curly}

Draw a Venn Diagram that correctly shows the relationship between sets A, B and U.

see solution





EXAMPLE 1.2.3

Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}
E = {x|x is an even number}
S = {1, 2, 3, 4}

Draw a Venn diagram that correctly shows the relationship between sets E, S and U.

see solution





SET OPERATIONS

Let U = {x|x is an English-language film}

Set A below contains the five best films according to the American Film Institute.
A = {Citizen Kane, Casablanca, The Godfather, Gone With the Wind, Lawrence of Arabia}

Set B below contains the five best films according to TV Guide.
B= {Casablanca, The Godfather Part 2, The Wizard of Oz, Citizen Kane, To Kill A Mockingbird}

Set C below contains the five most passionate films according to tthe American Film Institute.
C = {Gone With the Wind, Casablanca, West Side Story, An Affair To Remember, Roman Holiday}.

Notice that some films appear on more than one of these lists.

SET INTERSECTION AND SET UNION

Casablanca and Citizen Kane are the films that are simultaneously in sets A and B. We say that
{Casablanca, Citizen Kane} is the intersection of A and B.

This is denoted:

A intersection symbol B{Casablanca,Citizen Kane}



Likewise,

A intersection symbol C { Gone With the Wind, Casablanca}



In general, if S and T are sets then S intersect T =



The shaded region of the Venn diagram below corresponds to S intersect T

Venn diagram with circles S and T overlapping within rectangle U. The region in which the two cirles overlap is shaded.







Suppose we merge all of the elements of A with all of the elements of B to form a single, larger set:

{ Citizen Kane, Casablanca, The Godfather, Gone With the Wind, Lawrence of Arabia, The Godfather Part 2, The Wizard of Oz, To Kill A Mockingbird}



This new set is called the union of A with B, and is denoted: A union symbol B



List the elements of .



In general, for any sets S and T,

S union T = {x|x is an element of S or x is an element of T



The shaded region of the Venn diagram below corresponds to S union T

Venn diagram showing circles S and T overlapping within rectangle U. Both circles are shaded.





SET INTERSECTION, SET UNION, SET COMPLEMENT: SUMMARY

The intersection of two sets denotes the elements that the sets have in common, or the "overlap" of the two sets.





The union of two sets merges the two sets into one "larger" set.





The complement of a set consists of all elements in the universal set that are not in the given set.







Who's to blame?


EXAMPLE 1.2.4

For problems 1 - 10 refer to these sets:
U = {a, b, c, d, e, f}
A = {a, c, e, f}
B = {c, d, e}
C = {e, f}


Find each of the following:

1. A complement
2. B complement
3. C complement
4. B union C
5. A intersect C
6. B intersect C
7. (A union B) complement
8. A complement union B complement
9. B complement intersect C
10. A union (B complement intersect C)
see solutions for 1 - 10

World Wide Web note:
For more practice exercises involving set operations, visit Mr. Wooland's home page and try The BIG OPERATOR

Note: Problems 11-16 involve calculations that are explicitly visual in nature. For these problems ignore the images and shading. Instead, assume the following:
U = {1, 2, 3, 4}
A = {1, 2}
B = {2, 3}


11. A intersect B
12. A union B
13. A union B complement
14. A intersect B complement
15. (A union B) complement
16. A complement intersect B complement


Answers: 11. {2}
12. {1, 2, 3}
13. {1, 2, 4}
14. {1}
15. {4}
16. {4}


EXAMPLE 1.2.4A

Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
S = {2, 5, 7, 9}
V = {3, 4, 5, 6, 7}
T = {1, 3, 4, 5, 8, 9}

Find
(S complement intersect T) complement
see solution





DeMorgan's Laws for Set Mathematics

For any sets S, T:

(S union T) complement = S complement intersect T complement
"The complement of a union is the intersection of the complements."



(S intersect T) complement = S complement union T complement
"The complement of an intersection is the union of the complements."





Who's to blame?



EXAMPLE 1.2.5

The following exercises are explicitly visual in nature. Ignore any reference to graphic images and instead assume the following:
U = {1, 2, 3, 4, 5, 6, 7, 8}
A = {1, 2, 4, 5}
B = {2, 3, 5, 6}
C = {4, 5, 6, 7}





1. (A complement intersect B) intersect C complement
2. A intersect (B intersect C complement)
3. (A union B) intersect C
4. (A intersect C complment) union B complement
5. (A complment union B) complement intersect C
6. A complement union (B complement intersect C)


Answers
1. {3}
2. {2} 3. {4, 5, 6}
4. {1, 2, 4, 7, 8}
5. {4}
6. {3, 4, 6, 7, 8}
EXAMPLE 1.2.6 (A CLAST-style problem)
This example is explicitly visual in nature. Please omit it.
Sets A, B, C and U are related as shown in the diagram.

Which of the following statements is true, assuming that none of the regions is empty?

A. Any element that is a member of C is also a member of A and of B.
B. There is no element common to set A and set C.
C. Any element that is a member of set A is also a member of set B.
D. None of the above statements is true.



WORLD WIDE WEB NOTE:
For more examples like the previous one, visit Mr. Wooland's home page and try
MY OTHER COUSIN VENNY


Download practice exercises (PDF file)