Transcript Working WITH Sets
Working WITH Sets
Section 3-5
Goals
Goal
• To write sets and identify subsets.
• To find the complement of a set.
Rubric
Level 1 – Know the goals.
Level 2 – Fully understand the goals.
Level 3 – Use the goals to solve simple problems.
Level 4 – Use the goals to solve more advanced problems.
Level 5 – Adapts and applies the goals to different and more complex problems.
Vocabulary
• Set • Roster Form • Set-Builder Notation • Empty Set • Venn Diagram • Universal Set • Complement of a Set • Subset
Why Study Set Theory?
Understanding set theory helps people to … • see things in terms of systems • organize things into groups • begin to understand logic
Set Concepts
Studying sets helps us categorize information. It allows us to make sense of a large amount of information by breaking it down into smaller groups.
Sets:
•A
set
is a collection of objects.
–These objects can be anything: Letters, Shapes, People, Numbers, Desks, cars, etc.
–Notation: Braces ‘{ }’, denote “
The set of …
” •These objects are called
elements
or
members
of the set. •The symbol for element is .
•For example, if you define the set as all the fruit found in my refrigerator, then apple and orange would be elements or members of that set.
Sets:
• Sets are inherently
unordered
: – No matter what objects a, b, and c denote, {a, b, c} = {a, c, b} = {b, a, c} = {b, c, a} = {c, a, b} = {c, b, a}.
• All elements are
distinct
(unequal); multiple listings make no difference!
– {a, b, c} = {a, a, b, a, b, c, c, c, c}. – This set contains at most 3 elements!
Sets:
• • There are three methods used to indicate a set: 1. Description 2. Roster form 3. Set-builder notation
Venn Diagram
- Used to display the contents of a set and the relationships between sets.
1. Description:
•Description means just that, words describing what is included in a set. •Example: “ Set
M
is the set of months that start with the letter J.
”
2. Roster Form:
•Roster form lists all of the elements in the set within braces {element 1, element 2, …}.
•Example: Set
M
= { January, June, July}
3. Set-Builder Notation:
•Set-builder notation is frequently used in algebra. •Example: year and
x M
= {
x
x
is a month of the starts with the letter J} •This is read, “Set
M
is the set of all the elements
x
year and
x
such that
x
is a month of the starts with the letter J”.
Set Summary:
•In summary the three methods used to describe a set are: 1) Description: Set
A
is the integers 1, 2, 3, and 4.
2) Roster form: Set
A
= { 1, 2, 3, 4 } 3) Set-builder notation: –
A
= {
x
x
= 1, 2, 3, 4 }
Designating Sets
Sets are commonly given names (capital letters).
A
= {1, 2, 3, 4}
The set containing no elements is called the
empty set
(
null set
) and denoted by { } or .
The Empty Set
• Any set that contains no elements is called the
empty set
• the empty set is a subset of every set including itself • notation: { } or
Examples ~ both A and B are empty
A = {x | x is a Chevrolet Mustang} B = {x | x is a positive number 0}
Set Notation Elements
• an element is a member of a set • notation: means “is an element of” means “is not an element of” • Examples: – –
A = {1, 2, 3, 4}
1 2 A A 6 z A A
B = {x | x is an even number
2 B 9 B 4 B z B
10}
Example: Listing Elements of Sets
Give a complete listing of all of the elements of the set {
x
|
x
is a natural number between 3 and 8} Solution {4, 5, 6, 7} When listing the elements of a set, elements that occur more than once, are not repeated when listing the elements in set notation.
Set Theory Notation Summary
Symbol Meaning
Upper case Lower case { } or | or : designates set name designates set elements enclose elements in set is (or is not) an element of such that (if a condition is true)
VENN DIAGRAMS
Venn diagrams are useful for presenting a visual picture of set relationships.
Venn Diagrams
• Sets can be represented graphically using Venn diagrams.
• In Venn diagrams: –A rectangle represents the universal set.
–Circles (and other geometric figures) represents sets.
–Points (or words, nunbers) represent elements.
Venn Diagrams
Subsets
•When working with a large group of information, we often break it into smaller sets called
subsets
.
Subsets of a Set
Set
A
is a subset of set
B
if every element of
A
is also an element of
B.
this is written
A
B
.
In symbols
B A U
Subsets:
• Set
A
is a
subset
of set
B
, symbolized by
A
B
, if and only if all the elements of set
A
are also elements of set
B
. So to be a subset, all elements of the set are also elements in another set (which is either the same size or larger than the first set).
Subsets
• a subset part of or equal to another set • notation: means “is a subset of” means “is not a subset of”
Subset Examples:
• 1.
Given the sets 3 }, and
D A
B
(said “
A A
= { 1 , 2 }, = { 1 , 2 , 3 }
B
is a subset of = { 1 , 2 ,
B
”) since all
A
is in
B
. Note that this cannot be written in reverse since
B
is not a subset of
A
.
2. D
B
since all
D
is in
B
.
Example: Subsets
statement.
a) {a, b, c} ___ { a, c, d} b) {1, 2, 3, 4} ___ {1, 2, 3, 4}
Solution
Other Interesting Points About Subsets:
1. A
A
itself). (meaning - every set is a subset of 2. The empty set, , is a subset of every set, including itself.
Number of Subsets
The number of subsets of a set with
n
elements is 2
n
.
Example: Number of Subsets
Find the number of subsets of the set {m, a, t, h, y}.
Solution Since there are 5 elements, the number of subsets is 2 5 = 32.
One Last Point:
•The number of distinct subsets of a finite set
A
is 2
n
, where
n
is the number of elements in set.
•Example: Given the set { S,L,E,D } . The set has 4 elements, and 2 4 = 16. Thus, there are 16 distinct subsets for that set (note that the empty set is one of those 16 sets).
More Sets:
•Two more important sets to consider are the empty set (also called null set) and the universal set. •The
empty set
is the set that contains no elements. It is symbolized by { } or by . •The
universal set
, symbolized by
U
, is the set of all elements for any specific discussion.
Universal Set
• The universal set is the set of all things pertinent to a given discussion and is designated by the symbol
U Example : U
= {all students at ATC} Some Subsets : A = {all HS students} B = {freshmen students} C = {sophomore students}
Universal Set Example
• the universal set is a deck of ordinary playing cards • each card is an element in the universal set • some subsets – face cards are: – numbered cards – suits – poker hands
Universal Set
In a venn diagram the rectangle represents the universal set,
U
, and it is required for all venn diagrams.
A U
More on the Empty Set
• A set that has no elements is called the
empty set
or
null set
.
• Yes, it is still considered a real set, even though it has no elements.
• It is denoted by , or by { }.
• Since the empty set is a set, another set can contain the empty set as one of its elements: A ={ , a} This set has 2 elements B = { } This set has 1 element C = This set has 0 elements
Empty Set and Universal Set – Example: •If we are given the universal set •
U
= { Chris, Tom, Alex }, then only these three names can be considered when working with the problem. •If
A
= {
x
letter J}, then our answer would be the empty set (
x
U
and
x
starts with the ), since none of the names in our universal set start with the letter J.
Complement of a Set:
•The
complement of a set A
, symbolized by
A
, is all the elements in the universal set that are not in
A
(everything outside of
A
). One easy way to find this if the sets are in roster form is to cross out each element in
U
that is in set
A
. Then, whatever is not crossed out in
U
, is an element of
A
. If
U
= { 1 , 2 , 3 , 4 , 5 , 6 }, and then
A
A
Complement of a Set:
• The
complement
of a set is the set of elements which do not belong to the set being complemented.
• Equivalent to the logic operation “not” • • Written as a prime, A ’ , or a superscripted ‘c’, A c .
Example:
U = {a, b, c, d, e, u, v, w, x, y, z} A = {a, b, c, x, y, z} and B = {a, b, c, d, e} A ’ = {d, e, u, v, w} B
c
= {u, v, w, x, y, z}
Complement of Sets: Venn Diagrams
U = {a, b, c, d, e, u, v, w, x, y, z} A = {a, b, c, x, y, z} B = {d, e, y, z} A ’ = {d, e, u, v, w}
Joke Time
• What is the best state to buy a new soccer uniform in?
• New Jersey • Why is a football stadium always a cool place to sit?
• It’s full of fans!
• What did the pony say when he had a cold?
• I’m just a little horse!
Assignment
3-5 Exercises Pg. 213 – 215: #10 – 56 even