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