Foundations 120 Section 3.1

Types of Sets & Set Notation

Investigating Sets and Set Notation

Set: a collection of distinct objects. For example,

is the set of Natural numbers less than 7.

Element: each object in a set. For example, 4 is an element of N, the set of Natural numbers.

Universal Set: a set of all the elements under consideration for a particular context. For example:

is the universal set of all the students in Ms. Sewell’s Period 4 Foundation 12 class this semester.

Subset: a set whose elements all belong to another set. For example,

, the students in the front row is a subset of all the students in Ms.Sewell’s Period 4 Foundation 12 class this semester. In set notation, this relationship is written

Complement: all the elements of a universal set that do not belong to a subset of it. For example,

are all the students in Ms. Sewell’s Foundation class who do not sit in the front row. The complement is denoted with the prime sign, .

Empty Set: a set with no elements. For example, the set of odd numbers divisible by 2, is an empty set

as well as a subset of the odd numbers. The set of Natural numbers less than 1, is also an example of an

empty set. as well as a subset of the Natural numbers. You can think of an empty set as a box with nothing in it. The empty set is a subset of every set.

Disjoint: Two or more sets having no elements in common. For example, the set of even numbers and the set of odd numbers, the students in Ms. Sewell’s Foundation class and the students in Ms. Stewart’s Foundation class.

Erin is studying the provinces and territories of Canada. How can she use sets to categorize Canada’s regions?

1. List the elements of the universal set of Canadian provinces and territories, C.C = { BC, AB, SK, MB, ON, QC, NB, NS, PEI, NL, YT, NT, NU}

{ } or O O

Foundations 120 Section 3.1

2. One subset of C is the set of Western provinces and territories, W. Write W in set notation.W = { BC, AB, SK, MB, YT, NT, NU}Venn Diagram: a diagram representing mathematical or logical sets pictorially as circles or closed curves within an enclosing rectangle (the universal set), common elements of the sets being represented by the areas of overlap among the circles.

3. The Venn diagram represents the universal set, C. The circle in the Venn diagram represents the subset W. The complement of W is W’.

a. What does W’ contain?W’ is the set of Canadian provinces & territories that are not elements of W. They are the provinces of eastern Canada.

b. Write W’ in set notation.W” = {ON, QC, NB, NS, PEI, NL}

c. Explain what W’ represents in the Venn diagram.W’ represents all the elements in the universal set C that are not in W.

d. Erin wrote the set of Eastern provinces as . Is E equal to W’?

Yes, they are equal. They have exactly the same elements. (order does not matter)

e. List T, the set of territories in Canada. Is T a subset of C? Is it a subset of W, or a subset of W’. Explain.T ={YT, NW, NU} T is a subset of C and W, but not W’.

f. List B, the set of provinces east of Newfoundland.B = { } B is an empty set. There are no provinces east of NL.B =

g. Consider sets C, W, W’, and T. List a pair of disjoint sets. Is there more than one pair of disjoint sets?W and W’ Disjoint sets have no elements in common.W’ and T

h. Draw a Venn diagram illustrating sets C, W, W’ and T.

AB, BC, SK, MB, NT, NU, YT

ON, QC, NB, NS, PEI, NL

NT, YT, NU

BC, AB, SK, MB

T

T

Foundations 120 Section 3.1

Example 1: Sorting Numbers Using Set Notation and a Venn Diagram

a. Indicate the multiples of 4 and 12, from 1 to 60 inclusive, using set notation. List any subsets.

S is the universal set of all natural numbers from 1 to 60. There are too many to list as S = {1, 2, 3, …, 58, 59, 60}. Instead the shorthand method is used – set notation.

F = {4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60} F is the set of multiples of four from 1 to 60.(note 15 x 4 is the maximum value of 60)

T = {12, 24, 36, 48, 60} (note 12 x 5 is 60)

T is a subset of both F and S.F’ = non multiples of 4 from 1 to 60

b. Represent the sets and subsets in a Venn diagram. The numbers outside the 2 oval are F’.

SF

T

4, 8, 16, 20, 28, 32, 40, 44, 52, 56

12, 24, 36, 48

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 33, 34, 35, 37, 38, 39,41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59

Foundations 120 Section 3.1

Example 2: Determining the Number of Elements in Sets

Triangular numbers, such as 1, 3, 6, 10, or 15 can be represented as a triangular array.

a. Determine a pattern you can use to determine any triangular number.

b. Determine how many natural numbers from 1 to 100 are

even and triangular

First define the sets: U = {natural numbers from 1 to 100} T = { triangular numbers from 1 to 100} E = {even triangular numbers from 1 to 100}

1-100 means inclusive.n(X) means the number of elements in set X

T = { 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91} all triangular numbersn(T) = 13

E = { 6, 10,28, 36, 66, 78} n(E) = 6

Foundations 120 Section 3.1

odd and triangular

n(O) = n(T)-n(E) n(O) = 13 – 6n(O) = 7

not triangularn(U) = 100n(T’) = n(U)-n(T)n(T’) = 100-13n(T’) = 87

c. How many numbers are triangular?

n(T) is infinite

Infinite Set: a set with an infinite number of elements. The set of natural numbers would be an example of an infinite set.

Finite Set: a set with a countable number of elements. For example the set of natural numbers less than 7 is finite.

Foundations 120 Section 3.1

Example 3: Describing the Relationships Between Sets

The Toronto Zoo is currently featuring polar bears, western grey kangaroos, cheetahs, African lions, white pelicans, snowy owls, northern bald eagles, American alligators, and a boa constrictors.

a. Design a way to organize the animals for the zoo website. Represent your organization using a Venn diagram.

A = {all animals currently being featured at the zoo}W= {warm-blooded animals}W = {polar bears, kangaroos, cheetahs, lions, pelicans, snowy owls bald eagles}C = {cold blooded animals}C = { alligators, boa constrictors}M = {mammals}M = { polar bears, kangaroos, cheetahs, lions}B = {birds}B = { pelicans, snowy owls, bald eagles}

b. Name any disjoint sets. Remember these sets have no elements in common.

M & BW & CM & CB & C

c. Show which sets are subsets of one another using set notation.

For example:

CBM

W A

Alligators, boa constrictors

Snowy owls, pelicans, bald eagles

Polar bears, kangaroos, cheetahs, lions

Foundations 120 Section 3.1

Example 4: Solving a Problem Using a Venn Diagram

The possible sums that can occur when you roll two six-sided dice are recorded in the outcome table to the right.

a. Display the following sets in one Venn diagram:i. Rolls that produce a sum less than six

ii. Rolls that produce a sum greater than six.

S = {all possible sums}L = { all sums less than six}G = {all sums greater than 6}

b. Record the number of elements in each set.

n(S)=36 n(L) = 10 n(G) = 21

Sets L and G are disjoint – they are mutually exclusive

c. Determine a formula for the number of ways that a sum less than or greater than six can occur. Verify your formula.n(L or G)= n(L) +n(G) To verify: 36 total sumsN (L or G) =10 +21 5 ways the sum can be 6n(L or G) = 31 36-5 =m 31 31 ways the sum can be greater than 6 or less than 6

Mutually Exclusive: two or more events that cannot occur at the same time. For example:

When you toss a die once, it can be 1 or 2 or 3 or 4 or 5 or 6 but not all of them.

G(1,6), (2,5), (2,6), (3,4),(3,5), (3,6), (4,3), (4, 4), (4, 5), (4, 6), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1),(6,2),(6,3),(6,4),(6,5), (6,6)

L(1,1), (1,2), (1,3), (1,4), (2,1), (2,2),(2,3), (3,1), (3,2),((4,1)

S

Foundations 120 Section 3.1

Leaving the FHS parking lot after school, you can turn left or right but not both. A light cannot be red and green at the same time It cannot be 2:00 pm and 6:00pm in Fredericton right now I cannot be in Foundations 12 period 3 and Chemistry 12 period 3

Something not mutually exclusive: you can turn into your driveway & open the garage door at the same time.