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Chapter 7 Sets and Probability . Section 7.1 Sets. What is a Set? . A set is a well-defined collection of objects in which it is possible to determine whether or not a given object is included in the collection. Example: The letters of the Alphabet. The Vocabulary of Sets.

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what is a set
What is a Set?
  • A set is a well-defined collection of objects in which it is possible to determine whether or not a given object is included in the collection.
  • Example: The letters of the Alphabet
the vocabulary of sets
The Vocabulary of Sets
  • Each object in the set is referred to as an elementor member of the set. The symbol denotes membership in a set, while is used to show an element is notan element.
  • Example: S = { 2, 4, 6, 8, …}

12  S

25  S

  • It is possible to have a set with no elements. This kind of set is called an empty setand is written as { } or .
slide4
Equal sets have exactly the same elements.
  • Equivalent setshave the same number of elements.
  • Example: A = { d, o, g }

B = { c , a , t }

C = { d, o, g, s }

D = { a, c, t }

Which, if any, of the sets are equal?

B = D

Equivalent?

A, B, and D are equivalent.

Not equal?

C is not equal to any of the sets.

slide5
The cardinality of set A refers to the number of elements in set A and is written as n(A).
  • Example:Set Z is defined as containing all the single digits. List each element in the set, then find n(Z).

Z = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 }

Note: Listing each member of a set one time is called roster, or listing, notation.

n(z) = 10

slide6
Sometimes it is not convenient, or feasible, to list each element of a set.
  • When we are interested in a common property of the elements in a set, we use set-builder notation.

{ x| x has property P }

“The set of all x such that x has property P”.

  • Example: Use set-builder notation to write the set of elements defined as a number greater than 10.

{ x| x > 10 }

slide7
The universal setis a set that includes all objects being discussed.
  • Sometimes every element of one set also belongs to another set. This is an example of a subset.

B = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }

A = { 2, 4, 6, 8, 10 }

A is a subset of B

slide8
Set A is a subset of set B (written A  B) if every element of A is also an element of B.
  • Set A is a proper subset (written A B) if A  B and A  B.
  • The symbol  is used to describe an improper subset in which the subset and set are equal.
slide9
For any set A,   A and A  A .
  • Example: List all possible subsets of {x, y}.

There are 4 subsets of {x, y} :

, proper subset

{x}, proper subset

{y}, proper subset

{x, y} improper subset

  • A set of n distinct elements has subsets.
set operations
Set Operations

Given a set A and a universal set U, the set of all elements of U that do not belong to A is called the complementof set A.

slide11
Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

A = {2, 4, 7, 9}

B = {3, 5, 8, 10}

C = {1, 3, 5, 7, 9}

Find each of the following sets.

1.) A'

A' = {1, 3, 5, 6, 8, 10}

2.) B'

B' = {1, 2, 4, 6, 7, 9}

3.) C '

C '= {2, 4, 6, 8, 10}

slide12

Given two sets A and B, the set of all elements belonging to both set A and set B is called the

intersection of the two sets, written A B.

slide13
Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

A = {2, 4, 7, 9}

B = {3, 5, 8, 10}

C = {1, 3, 5, 7, 9}

Find each of the following sets.

1.) A  C

A  C = {7, 9}

2.) B  C

B  C = {3, 5}

3.) A  B

A  B = { } or 

disjoint sets
Disjoint Sets

For any sets A and B, if A and B are disjoint sets, then A B =  .

In other words, there are no elements that sets A and B have in common.

slide15

The set of all elements belonging to set A, to set B,

or to both sets is called the

union of the two sets, written A  B.

slide16
Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

A = {2, 4, 7, 9}

B = {3, 5, 8, 10}

C = {1, 3, 5, 7, 9}

Find each of the following sets.

1.) A  C

A C = {1, 2, 3, 4, 5, 7, 9}

2.) B  C

B C = {1, 3, 5, 7, 8, 9, 10}

3.) A  B

A B = {2, 3, 4, 5, 7, 8, 9, 10}