Set Operations

MATH 102 Contemporary Math S. Rook

Overview

• Section 2.3 in the textbook: – Intersection & Union – Complement – Difference

Intersection & Union

Intersection of Sets

• Given sets A and B, the

intersection elements common to both sets

of A and B, denoted , means to

list those

– Only those elements present in BOTH A and B are part of the

intersection

– e.g. Let A = {1, 2, 3, 4, 5} and B = {2, 4, 6, 8}. Find

A



B

– Can also represent using a

Venn Diagram

Union of Sets

• Given sets A and B, the

union

denoted , means to

elements of A and B together

of A and B,

combine the

– i.e. fill an initially empty set (bag) with the elements of A and then the elements of B – An element present in BOTH A and B is only added once to the

union

– e.g. Let A = {1, 2, 3, 4, 5} and B = {2, 4, 6, 8}. Find A U B – Can also represent using a

Venn Diagram

Union of Sets (Continued)

• Given sets A and B, what is the relationship between n(A), n(B), and n(A U B)?

– e.g. Let A = {p| p is a person sitting in the front row} and B = {g | g is a person wearing glasses}. What are the values of n(A) and n(B)?

• What is the value of n(A U B)?

n

•  Why does the sum of n(A) and n(B) not equal n(A U B)?

A



B

 

n

(

A

) 

n

(

B

) 

n



A



B

 – Verify for yourself that the value of n(A U B) checks for sets A and B on the previous slide

Intersection & Union (Example)

Ex 1:

Let A = {2, 3, 4, 5, 7, 9}, B = {x | x is an even natural number}, and C = {y | y is an odd natural number} . Find the following sets: a) b) c)

A



B B



C B



C

Complement

Complement

• •

Universal set

: the set of all elements being considered in a problem. Often denoted by U.

– All subsets in a problem are taken from the

universal set

Given set A, the means to list those elements that A is missing from the

universal set complement

of A, denoted by A’, – i.e. those elements that need to be added to A to

complete

the

universal set

– – e.g. Let U = {1, 2, 3, … , 10} and A = {1, 6, 9, 10}. Find A’.

Can also represent using a

Venn Diagram

Complement (Example)

Ex 2:

Let U = {a, b, c, d, e, f, g, h}, A = {a, b, c, e, g}, B = {a, b, c, d, e, f, g, h}, and C = { }. Find: a) A’ b) B’ c) C’

Difference

Difference

• Given sets A and B, the

difference

of A and B, denoted A – B, means

the resulting set when the elements of B are removed from the elements of A

– i.e. Just like subtraction, we are taking away those elements in B away from A – e.g. Let A = {1, 2, 3, 4, 5, 6} and B = {2, 3, 4}. Find A – B.

– Can also represent using a

Venn Diagram

Difference (Example)

Ex 3:

Let U = {1, 2, 3, … }, A = {1, 3, 5, 7, 9, … }, B = {1, 2, 3, 4, 5, 6, 8}, and C = {2, 4, 6, 8}. Find the resulting set: a) B – C b) C – B c) A – C

Combining Set Operations (Example)

Ex 4:

Let U = {1, 2, 3, …, 10 }, A = {1, 3, 5, 7, 9}, B = {1, 2, 3, 4, 5, 6}, and C = {2, 4, 6, 7, 8}. Find the resulting set: a) b) 

A A

'  

B

 

B

 

C C

 '  '  '

Combining Set Operations (Example)

Ex 5:

Shade the appropriate regions in a Venn Diagram to represent the resulting set: a) b)  

A A

 

B C

  

C B



C

 '

Summary

• • • After studying these slides, you should know how to do the following: – Find the intersection and union of sets – Calculate the number of elements in the union of sets – – Find the complement and difference of sets Apply multiple set operations – Use Venn Diagrams to illustrate set operations Additional Practice: – See the list of suggested problems for 2.3

Next Lesson: – Survey Problems (Section 2.4)