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A SET is a collection of numbers, objects or people.
Each item in a set is referred to as an element or a
member.
Examples: - set of books for Math 436
- set of tools used by a carpenter
- set of furniture for a bedroom
416
- set of students that have completed Math
- set of people who were born during the
month of October
- set of people who work at a restaurant
- set of odd numbers
- set of numbers that are less than 6
There are several ways in which a SET can be expressed.
Three ways that will be used in this chapter are:
LISTING
SET-BUILDER NOTATION
VENN DIAGRAM
When describing a set, special brackets are used to
contain the set.
{ }
It is inappropriate to use either of the following
sets of brackets to represent a set.
( )
[ ]
LISTING or Roster form
- when representing a set using this form there are
rules to follow.
Set name must be a single uppercase letter like
A, B, X or N but NOT a, b, x, Am, or Bb
Each element must be separated by a comma or a
semicolon.
Each element must be present only once.
A specific order is not required however it is
often convenient to list terms in order.
Suspension points (…) may be used to replace
elements when there is a predictable sequence of
elements.
Example of some sets by LISTING or Roster form
P = {Nova Scotia, New Brunswick, Newfoundland and Labrador, Prince Edward Island}
A = {Tiger Woods, Roger Federer, Thierry Henry}
C = {Blue, White, Red}
n(A) = 3
n(C) = 3
T = {Teddy, Mihaela, Jean, Brian, Jack, Augusta, Vicky, Mariette}
n(T) = 8
M = {January, March, May, July, August, October, December} n(M) = 7
O = {1, 3, 5, 7, 9, …} n(O) = N/A
E={ }
S = {0, 1, 4, 9, 16, …} n(S) = N/A
B = {-2, -1, 0, 1, 2, … , 6} n(B) = 9
L = {1, 4, 3, 0, 2} n(L) = 5
D = {-40, -38, -36, … , 26, 28, 30}
n(E) = 0
n(D) = 36
Some of these sets are finite (countable number of elements) and
some are infinite (uncountable number of elements). The number
of elements in a set is referred to as the cardinal number of a
set. If a set is finite, it has a cardinal number. If it is infinite, it
does not. The cardinal number is denoted as shown n(set name).
For the first set above it would be shown as follows:
n(P) = 4
Notice that the cardinal number for set E is 0. This set is
a special set in that it has no elements. We have several
names for it – Empty Set or Null Set or Void Set. Either
way it has no elements and it can be represented in either
of 2 ways.
{ } or Ø
But not { Ø }
For our purposes we will be working with sets involving
numbers more than people or other things.
There are special sets involving numbers that you need
to know as they will be referred to often throughout
this and other Math courses.
See handout on Groups of Numbers.
SET-BUILDER NOTATION
- This is basically a form where the set is described
instead of listed. The description however follows a
strict format.
A set-builder description of the set of natural numbers
less than 10 is shown below.
A = { x Є N | x < 10}
As with listing sets the set name is a single uppercase
letter ( ‘A’ in this case).
When expressing with set-builder notation there are new
symbols and concepts that you must become familiar with.
Membership symbol, Є - this symbol means ‘is an element
of’ OR ‘belongs to’.
1ЄL
5 Є L 5 does not belong to L
L = {1, 4, 3, 0, 2}
3ЄL
0ЄL
-1 Є L -1 is not an element of L
A = { x Є N | x < 10}
When expressing in set-builder notation, the first section
inside the bracket states that ‘x Є ‘ of some universe. A
universe is a set from which all possible elements are
considered for the set being described. The universe is
usually natural numbers (N), integers (Z) or real numbers
(R).
Separating the first section in the brackets from the
second section, is a the following symbol ‘ | ’. This symbol
can be read as ‘such that’ OR ‘whereas’.
The second section in the brackets describes the elements
of the set. It basically restricts the elements from the
universe that satisfies the set.
N = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, …}
A = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
P = {Nova Scotia, New Brunswick, Newfoundland and Labrador, Prince Edward Island}
P = {x Є a province of Canada| x is an Atlantic province}
C = {Blue, White, Red}
C = {x Є a color| x is a color in the Montreal Canadiens jersey}
T = {Teddy, Mihaela, Jean, Brian, Jack, Augusta, Vicky, Mariette}
T = {x Є a PACC teacher| x is a Math teacher}
M = {January, March, May, July, August, October, December}
M = {x Є a month of the year| x has 31 days}
O = {1, 3, 5, 7, 9, …}
O = {x Є N| x is an odd number}
S = {0, 1, 4, 9, 16, …}
S = {x Є N| x is a perfect square}
B = {-2, -1, 0, 1, 2, … , 6}
B = {x Є Z| -3 < x < 7}
D = {-40, -38, -36, … , 26, 28, 30}
D = {x Є Z| x is an even number and -41 < x < 31}
The same set can be represented either by listing or set-builder
notation and if one form is presented, we should be able to convert it
to the other form.
VENN DIAGRAM – A closed figure containing the
elements of the set.
A = {x Є Z| -4 < x < 6 and x is an even number} … Set-builder notation
A = {-2, 0, 2, 4} … Roster form
Venn diagram
When constructing a Venn diagram, a closed
A
2
figure must be drawn with each element
represented inside by a dot. The name (A in
-2
4
this case) must sit outside of the closed
figure.
0
U = {0, 1, 2, 3, 4, 5, 6, 7}
A = {0, 1, 2, 3}
B = {3, 4, 5}
U
A
2
4
3
1
0
B
5
6
7
N
A = {x Є N| 2 < x < 7}
A
B = {x Є N| x = 3}
0
6
4
B
5
3
1
2
7
8
9
…
U
A
1
2
7
3
5
6
4
8
C
B
U={
A={
B={
C={
}
}
}
}
R
Q
Q’
Z
N
A = {x Є N| 4 < x < 7}
4
5
6
7
8
4
What is the difference
What is the difference
What is the difference
What is the difference
C = {x Є R| 4 < x < 7}
4
5
6
B = {x Є N| 5 ≤ x ≤ 6}
7
8
between
between
between
between
5
6
7
8
sets A and B?
sets A and C?
sets B and D?
sets C and D?
D = {x Є R| 5 ≤ x ≤ 6}
4
5
6
7
8