I can provide examples of the empty set, disjoint sets,
subsets, and universal sets in context, and explain the reasoning.
I can organize information such as collected data and number
properties, using graphic organizers, and explain the reasoning. I
can determine the elements in the complement of two sets.
Slide 3
Explore Consider the following diagram that looks at one
possible way of grouping the letters of the alphabet.
Slide 4
Explore 1. How are the letters grouped? 2. Why is the letter Y
in an overlapping section of the diagram? 3. How many letters are
in the first circle? How many letters are in the second circle?
When you add these two numbers together, is the answer the same as
the number of letters in the alphabet? Why or why not? 4. Are there
any other ways that you could group these letters? Try this explore
activity in your workbook before looking at the answers on the next
slide.
Slide 5
You should notice 1. How are the letters grouped? They are
grouped according to their classification of vowel or consonant. 2.
Why is the letter Y in an overlapping section of the diagram? Y is
considered to be both a consonant and a vowel. 3. How many letters
are in the first circle? How many letters are in the second circle?
When you add these two numbers together, is the answer the same as
the number of letters in the alphabet? Why or why not? There are 6
items in circle A and 21 in circle B. This adds to 27, since Y is
counted twice. 4. Are there any other ways that you could group
these letters? Yes. For example, according to whether or not the
letter contains a straight line.
Slide 6
Information A set is a collection of distinguishable objects.
The objects in a set are called elements. A universal set contains
all the elements in a particular context. It is also called the
sample space. For example, the universal set of digits is D = {0,
1, 2, 3, 4, 5, 6, 7, 8, 9}. A subset is a smaller set whose
elements all belong to a bigger set. The complement is the leftover
elements of a universal set that do not belong to any subset.
Slide 7
Information Disjoint sets are two or more sets that have no
elements in common. An empty set is a set that has no elements in
it. A Venn diagram is a graphical way to show all the possible
relations between a number of sets. Numbers in a Venn diagram can
represent the number of elements in a particular set, or be the
numbers themselves.
Slide 8
Example 1 Canadas provinces and territories can be categorized
using sets. Using sets to categorize items
Slide 9
Example 1 a) List the elements of the universal set, C, of
Canadian provinces and territories. Sets are defined using
brackets. For example: U = {1, 2, 3} C = {BC, AB, SK, MT, ON, QC,
NL, NS, NB, PEI, YU, NWT, NVT}
Slide 11
Example 1 c) List T, the set of territories. Is T a subset of
another set? T = {YU, NVT, NWT} T, the Canadian territories is a
subset of W and also a subset of C. This is because all of the
territories in Canada are considered to be in the subset W.
Slide 12
Example 1 d) The complement W is W. i. Describe what W contains
ii. Write W in set notation. iii. Explain what represents in the
Venn diagram from part c). i. W contains all Canadian provinces and
territories that are not considered to be western. ii. W = {ON, QC,
NL, PEI, NS, NB} iii. W is all the elements in C that are not in
W.
Slide 13
Example 1 e) The set of eastern provinces is E = {NL, PEI, NS,
NB, QC, ON}. Is E equal to W? Explain. Yes. E is equal to W, since
they contain all of the same elements.
Slide 14
Example 1 f) Draw a Venn diagram representing C, W, T, and E.
List all elements in the appropriate circle(s). C W BC SK NVT AB YU
MT NWT T E NL PEI NS NB QC ON
Slide 15
Example 1 g) Explain why you can represent the set of Canadian
provinces south of Mexico by the empty set. There are no Canadian
provinces South of Mexico. The empty set is written as { } or . h)
Consider the sets C, W, E, W, and T. List all the pairs of disjoint
sets. Disjoint sets include W and E, W and W, T and E, T and W.
Canada Mexico
Slide 16
Example 2 A triangular number, such as 1, 3, 6, or 10, can be
represented as a triangular array. a) Determine a pattern you can
use to find any triangular number. Determining the number of
elements in sets +2 +4 +3 For each new triangle, add a number 1
more than the previous added number.
Slide 17
Example 2 b) Determine the following sets: i. U = {natural
numbers from 1 to 21 inclusive} ii. T = {triangular numbers from 1
to 21 inclusive} iii. E = {even triangular numbers from 1 to 21
inclusive} iv. O = {odd triangular numbers from 1 to 21 inclusive}
v. T = {non-triangular numbers from 1 to 21 inclusive} Before you
can do all of this, continue the pattern to determine all
subsequent triangular numbers: 1, 3, 6, 10, 15, 21 Try this example
in your workbook before looking at the answers on the next
slide.
Slide 18
Example 2 b) i. U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13,
14, 15, 16, 17, 18, 19, 20, 21} ii. T = {1, 3, 6, 10, 15, 21} iii.
E = {6, 10} iv. O = {1, 3, 15, 21} v. T = {2, 4, 5, 7, 8, 9, 11,
12, 13, 14, 16, 17, 18, 19, 20} 1, 3, 6, 10, 15, 21
Slide 19
Example 2 c) Using set notation, how many elements are in sets
U, T, and E? d) Using set notation and your answers from part c),
calculate the number of elements in sets O and T. In set notation,
the number of elements of set X is written as n(x). n(U) = 21n(T) =
6n(E) = 2 n(O) = 6 2 = 4 n(T) = 21 6 = 15
Slide 20
Example 2 e) Draw a Venn diagram to represent the sets in (b).
O T U E T
Slide 21
Example 2 f) Using set notation, list the sets that are subsets
of other sets. O T U E T
Slide 22
Example 2 g) Why is the set of odd triangular numbers from 1 to
21 defined as set O instead of being defined as E? h) Consider the
set of all natural numbers. How many numbers are triangular
numbers: Is this set finite or infinite? An infinite set, such as
the set of natural numbers, is written as N = {1, 2, 3}. E would
include all natural numbers between 1 and 21 that are NOT even and
triangular. O includes only the number from 1 to 21 that are odd
triangular numbers. The triangular numbers continue on forever, so
it would be considered an infinite set.
Slide 23
Example 3 Jarrod and Luke rescue homeless animals. They
currently have cats, dogs, rabbits, hamsters, parrots, lovebirds,
iguanas, and snakes. The boys need to design a web page to help
them advertise to find homes for the animals. They first must
organize the animals, so they grouped them into the following sets:
A = {all the animals that are available} W = {warm-blooded animals}
C = {cold-blooded animals} Describing the relationships between
sets
Slide 24
Example 3 a) Using set notation, list the elements of W and C.
b) Two possible subsets of W are M, the set of mammals, and B, the
set of birds. Using set notation, list the elements of these
subsets. c) Use a Venn diagram to represent A, W, C, and the two
subsets of W. d) Name any disjoint sets. e) Use set notation to
show which sets are subsets of one another. f) Jarrod said that the
set of fur-bearing animals could form one subset. Name another set
of animals that is equal to this subset. g) How else might you
categorize the animals into sets and subsets? Try this example in
your workbook before looking at the answers on the next slide.
Slide 25
Example 3: Solution a) Using set notation, list the elements of
W and C. b) Two possible subsets of W are M, the set of mammals,
and B, the set of birds. Using set notation, list the elements of
these subsets. W = {cats, dogs, rabbits, hamsters, parrots,
lovebirds} C ={iguanas, snakes} M = {cats, dogs, rabbits, hamsters}
B ={parrots, lovebirds}
Slide 26
Example 3: Solution c) Use a Venn diagram to represent A, W, C,
and the two subsets of W. B W A M C
Slide 27
Example 3: Solution d) Name any disjoint sets. e) Use set
notation to show which sets are subsets of one another. f) Jarrod
said that the set of fur-bearing animals could form one subset.
Name another set of animals that is equal to this subset. g) How
else might you categorize the animals into sets and subsets? M and
B, M and C, B and C, W and C mammals Reptiles, birds and mammals,
nocturnal and not nocturnal.
Slide 28
Need to Know A set is a collection of distinguishable objects
called elements. A set can be represented by: listing the elements
in set notation; for example, A = {1, 2, 3, 4, 5} using words, for
example, A = {all integers greater than 0 and less than 6} In set
notation, the number of elements of set X is written as: n(X).
Slide 29
Need to Know A universal set, also known as the sample space,
contains all the elements in a particular context. A universal set
can be split into subsets, often in more than one way. For example,
if set A is a subset of set U, the notation is used. Venn diagrams
can be used to show how sets and their subsets are related. An
empty set is a set that has no elements in it and is denoted as {
}.
Slide 30
Need to Know Disjoint sets are two or more sets that have no
elements in common. The complement of a set contains all elements
that do not belong to the set. The sum of the number of elements in
a set and its complement is equal to the number of elements in the
universal set n(A)+n(A) = n(U). Youre ready! Try the homework from
this section.