Survey of Mathfaculty.niacc.edu/rodney-zehr/wp-content/uploads/sites/13/2014/03/SOM... · hair....
Transcript of Survey of Mathfaculty.niacc.edu/rodney-zehr/wp-content/uploads/sites/13/2014/03/SOM... · hair....
Survey of Math Mastery Math
Unit 6: Sets
Sets – Part 1
• Set – a well defined collection of objects;
The set of all grades you can earn in this class
The set of tools in a tool box
The set of holiday dishes
The set of natural numbers
The set of people that do not have a middle
name.
Important!!!
There are many new terms and symbols
being used in this section.
To be successful with this information, you must
be able to read the language of math and be
able to understand what I’m asking you to find.
Practice talking (aloud if needed) the questions
that are being asked.
Sets
• Elements (members) – values that belong
to a set; use the symbol
Describe the following:
hammer {screwdriver, hammer, nail}
2 {0, 1, 2, 3, 4 …}
8 {1, 3, 5, 7, 9}
Sets
• Elements (members) – values that belong
to a set; use the symbol
hammer {screwdriver, hammer, nail} means: hammer is a member of the set {screwdriver, hammer, nail}
2 {0, 1, 2, 3, 4 …} means: 2 is a member of the set {0, 1, 2, 3, 4 …}
8 {1, 3, 5, 7, 9} means: 8 is not a member of the set {1, 3, 5, 7, 9}
Sets
• Empty Set (Null Set) – the set that
contains no elements
– use the symbol { } or but NOT {}
Example: Give the set of states that
begin with the letter z. Since there are no states that begin with the letter z,
there are no elements in the set
Answer = { }
Sets
0 represents a number
represents a set with no elements
{0} represents a set with one element, 0
Think of the { } as a bag, so { 0 } could mean
“a bag with the number 0 in it”
{} represents a set with one element,
“a bag with nothing in it”
Sets
• Finite Set – a set that can be measured or counted
Example: {1, 2, 3, 4, 5} There are 5 elements in the set, therefore the set if finite.
• Infinite Set – a set that can not be measured
Example: {1, 2, 3, 4, 5 …} The ellipse (…) shows that the list keeps going without end, so the list
is not countable and therefore is an infinite list.
Sets Set Notation:
• The elements of the set are enclosed by brackets { } such as {Iowa, Illinois, Idaho, Indiana}
• The name of the set uses a capital letter
• An equal sign follows the name of the set such as I = {Iowa, Illinois, Idaho, Indiana}
• Letters used as elements need to be lower case such as A = {a, b, c, d, e}
Sets
Well-defined sets – a set that is specific
Are the following well-defined sets:
1. The set of letters grades students can
earn in a class
2. The set of all nice teachers at NIACC
Sets
Well-defined sets – a set that is specific
Are the following well-defined sets:
1. The set of letters grades students can
earn in a class Yes, because the set contains {A, B, C, D, F} as grades
2. The set of all nice teachers at NIACC No, because “nice teachers” is subjective (not specific) to everyone
Sets
Equal sets – sets that contains the exact
same elements
True or False: {a, b, c, d} = {a, d, b, c}
Sets
Equal sets – sets that contains the exact
same elements
True or False: {a, b, c, d} = {a, d, b, c}
True - the sets contains the exact same
elements just in a different order
Sets (Part 2)
Subset – when every element of the first set is also a member of the second set
*use the symbol
Using A = {1, 2} and B = {1, 2, 3, 4, 5}
a) True or False: A B
b) True or False: B A
Sets (Part 2)
Subset – when every element of the first set is also
a member of the second set
*use the symbol
Using A = {1, 2} and B = {1, 2, 3, 4, 5}
a) True or False: A B
True - because both elements of set A are in set B
b) True or False: B A False - because only 2 of the elements in set B are found in set A
Sets
Property:
The empty set is a subset of every set:
If C = {2, 4, 6}, is C?
Sets
The empty set is a subset of every set:
If C = {2, 4, 6}, is C?
Yes because the empty set is a subset of every set
OR
Yes because the empty set is “part” of every set
Sets
Property:
Every set is a subset of itself
If A = {cat, dog, rabbit}, is A A?
Sets
Every set is a subset of itself
If A = {cat, dog, rabbit}, is A A?
Yes, because every set is a subset of itself
OR
Yes, because every set is “part” of itself
Sets
Make a list of the subsets for {a, b, c}.
Sets
Make a list of the subsets for {a, b, c}.
{a} {b} {c}
{a, b} {a, c} {b, c}
{a, b, c}
{ }
Sets
Make a list of the subsets for {a, b, c}.
{a} {b} {c}
{a, b} {a, c} {b, c}
{a, b, c}
{ }
The number of subsets can be found by using 2k where k = the number of elements in the set
The number elements in {a, b, c} is three, so the number of subsets can be found by using 23 = 8 subsets.
Sets
Find the number of subset in
a) {a, e, i, o, u}?
b) a set containing 12 elements?
Sets
Find the number of subset in
a) {a, e, i, o, u}? 25 = 32 subsets
b) a set containing 12 elements?
212 = 4096 subsets
Sets (Part 3)
Universal set – includes all the objects being
discussed
Survey of Math is one of the courses in your
schedule; your schedule would represent
the universal set.
Venn Diagrams
U
The rectangle represents the universal set – represents the
set of all elements under consideration.
Creating a Venn diagram
1. Make a list of traits (elements) for each item:
Milk Coffee
Creating a Venn diagram
1. Make a list of traits (elements) for each item:
Milk Coffee White Brown
Cold Hot or cold
Wet Wet
Used on cereal Specialty shops
Comes from cows Made from beans
2. Create a Venn diagram for your items
Milk Coffee
White Brown
Cold Hot or cold
Wet Wet
Used on cereal Specialty items
Comes from cows Made from beans
U
2. Create a Venn diagram for your items
Milk Coffee
White Brown
Cold Hot or cold
Wet Wet
Used on cereal Specialty shops
Comes from cows Made from beans
U
M C
white
cold
wet
Use on cereal
Comes from cows
brown
Specialty shops
Made from beans
2. Create a Venn diagram for your items
Since there were traits in common (cold & wet), those elements need to be in the
overlap (intersection)
There can also be traits (elements) outside the circle, but within the rectangle. An
example for this example might be the trait “can be cut with a knife”
U
M C
white
cold
wet
Use on cereal
Comes from cows
brown
Specialty shops
Made from beans
Can be cut
with a knife
Intersection of two sets: A B
U
A
A B is presented by the shaded region
(the common elements shared by both sets)
B
Disjoint sets
U
A
A B = { }
Disjoint sets share no common elements
B
Union of two sets: A B
U
A
A B is represented by the shaded region; the
union of sets brings all elements together
B
Subsets
U
A
A B (all elements of set A are in set B)
B
EX1: Use the following information:
Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10,11}
A = {1, 2, 4, 5, 6}
B = {2, 4, 5, 7, 9, 11}
a) Draw a Venn diagram to represent the
above information
b) Find A B
c) Find A B
U
A B
1
6
2
4
5
7
9
11
3
10
8
U
A B
1
6
2
4
5
7
9
11
3
10
8
b) A B = {2, 4, 5}
c) A B = {1, 2, 4, 5, 6, 7, 9, 11}
EX2: Use the following information:
Let U = {0, 1, 2, 3, … , 12}
A = {1, 3, 5, 7, 9, 11}
B = {3, 6, 9, 12}
C = {1, 2, 3, 4, 5}
a) Draw a Venn diagram to represent the above information
b) Find A B
c) Find (A B) C
U
A B
C
7
11
9
3 5
1
6
12
2 4 8
10
b) A B = {1, 3, 5, 7, 9, 11}
c) (A B) C = {1, 3, 5} {1, 3, 5, 6, 7, 9, 11, 12} {1, 2, 3, 4, 5} = {1, 3, 5}
0
EX3: Twenty-four dogs are in a kennel. Twelve of the dogs are black, six of the dogs have short tails, and fifteen of the dogs have long hair. There is only one dog that is black with a short tail and long hair. Two of the dogs are black with short tails and do not have long hair. Two of the dogs have short tails and long hair but are not black. If all of the dogs in the kennel have at least one of the mentioned characteristics, how many dogs are black with long hair but do not have short tails?
Organize the information
24 total dogs (the universal set contains 24 elements)
12 are black B = set of dogs with black hair
6 have short tails S = set of dogs with short tails
15 have long hair L = set of dogs with long hair
1 is black with long hair and a short tail (1 has all three characteristics)
2 are black with short tails
2 have short tails and long hair
All dogs have at least one of the characteristics
Create a Venn diagram to help
solve the problem: U
B
S L
Start with intersection of all three traits (1)
U B
S L
1
Move to two traits: 2 black with short tails (the football shape needs to equal 2)
U B
S L
1
1
Move to two traits: 2 have a short tail & long hair (the football shape needs to equal 2)
U B
S L
1
1
1
Move to one trait: 6 have long hair (circle for set S must equal 6)
U B
S L
1
1
1
3
Move to one trait: 15 have short tails (circle for set S must equal 6) but we are missing
part of the information (x)
U B
S L
1
1
1
3
x
13 - x
Move to one trait: 12 are black (circle for set B must equal 12) but we are missing part of
the information (x)
U B
S L
1
1
1
3
x
13 - x
10 - x
Solve for x: the sum of each section of the Venn diagram = 24
U B
S L
1
1
1
3
x
13 - x
10 - x
10 – x + 13 – x + 3 + 1 + 1 + 1 + x = 24
29 – x = 24
x = 5 dogs that are black and have long tails
U B
S L
1
1
1
3
x
13 - x
10 - x
Time to practice!!
• Understand all symbols
• Read the language of math
• Organize information in a Venn diagram
• Use the data in a Venn diagram