Theory of the sets and Venn diagram-grade 9 math

sets

1. Unit 1 - Sets
Grade 9 - 2025
1

2. What is a set?
•A set is a group of “objects”
•People in a class: { Alice, Bob, Chris }
•Classes offered by a department: { CS 101, CS 202, … }
•Colors of a rainbow: { red, orange, yellow, green, blue, purple }
•States of matter { solid, liquid, gas, plasma }
•States in the US: { Alabama, Alaska, Virginia, … }
•Sets can contain non-related elements: { 3, a, red, Virginia }
•Although a set can contain (almost) anything, we will most often use
sets of numbers
•All positive numbers less than or equal to 5: {1, 2, 3, 4, 5}
•A few selected real numbers: { 2.1, π, 0, -6.32, e } 2

3. Set properties 1
•Order does not matter
•We often write them in order because it is easier for
humans to understand it that way
•{1, 2, 3, 4, 5} is equivalent to {3, 5, 2, 4, 1}
•Sets are notated with curly brackets
3

4. Set properties 2
•Sets do not have duplicate elements
•Consider the set of vowels in the alphabet.
•It makes no sense to list them as {a, a, a, e, i, o, o, o, o, o, u}
•What we really want is just {a, e, i, o, u}
•Consider the list of students in this class
•Again, it does not make sense to list somebody twice
•Note that a list is like a set, but order does matter and
duplicate elements are allowed
•We won’t be studying lists much in this class
4

5. Specifying a set 1
•Sets are usually represented by a capital letter (A, B, S,
etc.)
•Elements are usually represented by an italic lower-case
letter (a, x, y, etc.)
•Easiest way to specify a set is to list all the elements: A
= {1, 2, 3, 4, 5}
•Not always feasible for large or infinite sets
5

6. Specifying a set 2
•Can use an ellipsis (…): B = {0, 1, 2, 3, …}
•Can cause confusion. Consider the set C = {3, 5, 7, …}. What
comes next?
•If the set is all odd integers greater than 2, it is 9
•If the set is all prime numbers greater than 2, it is 11
•Can use set-builder notation
•D = {x | x is prime and x > 2}
•E = {x | x is odd and x > 2}
•The vertical bar means “such that”
•Thus, set D is read (in English) as: “all elements x such that x is
prime and x is greater than 2”
6

7. 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’.
L = {1, 4, 3, 0, 2}
1 Є L
3 Є L
0 Є L
5 Є L
-1 Є L
5 does not belong to L
-1 is not an element of L

8. N = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, …}
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.
A = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

9. C = {Blue, White, Red}
T = {Teddy, Mihaela, Jean, Brian, Jack, Augusta, Vicky, Mariette}
O = {1, 3, 5, 7, 9, …}
S = {0, 1, 4, 9, 16, …}
M = {January, March, May, July, August, October, December}
C = {x Є a color| x is a color in the Montreal Canadiens jersey}
T = {x Є a PACC teacher| x is a Math teacher}
M = {x Є a month of the year| x has 31 days}
O = {x Є N| x is an odd number}
B = {-2, -1, 0, 1, 2, … , 6}
S = {x Є N| x is a perfect square}
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.

10. Specifying a set 3
•A set is said to “contain” the various “members” or
“elements” that make up the set
•If an element a is a member of (or an element of) a set S,
we use then notation a  S
•4  {1, 2, 3, 4}
•If an element is not a member of (or an element of) a set
S, we use the notation a  S
•7  {1, 2, 3, 4}
•Virginia  {1, 2, 3, 4}
10

11. Often used sets
•N = {0, 1, 2, 3, …} is the set of natural numbers
•Z = {…, -2, -1, 0, 1, 2, …} is the set of integers
•Z
+
= {1, 2, 3, …} is the set of positive integers (a.k.a
whole numbers)
•Note that people disagree on the exact definitions of whole
numbers and natural numbers
•Q = {p/q | p  Z, q  Z, q ≠ 0} is the set of rational
numbers
•Any number that can be expressed as a fraction of two
integers (where the bottom one is not zero)
•R is the set of real numbers
11

12. The Cardinality of a Set
•Notation: n(A)
•For finite sets A, n(A) is the number of elements of
A.
•For infinite sets A, write n(A)=∞.

13. Universal Sets and Venn Diagrams
•The universal set is a general set that contains all elements under discussion.
•John Venn (1843 – 1923) created Venn diagrams to show the visual
relationship among sets.
•Universal set is represented by a rectangle
•Subsets within the universal set are depicted by circles, or sometimes ovals or
other shapes.
03/10/25Section 2.3 13

14. Venn diagrams
•Represents sets graphically
•The box represents the universal set
•Circles represent the set(s)
•Consider set S, which is
the set of all vowels in the
alphabet
•The individual elements
are usually not written
in a Venn diagram
14
a
e i
o u
bcdf
ghj
klm
npq
rst
vwx
yz
U
S

15. Example 1
Determining Sets From a Venn Diagram
•Use the Venn diagram to determine each of the following sets:
a.U
U = { O ,
∆
, $, M, 5 }
b.A
A = { O ,
∆
}
c.The set of elements in U that are not in A.
{$, M, 5 }
03/10/25Section 2.3 15

16. Representing Two Sets in a Venn Diagram
Disjoint Sets: Two sets that haveEqual Sets: If A = B then AB
no elements in common.and B  A.
Proper Subsets: All elements of Sets with Some Common Elements
set A are elements of set B. Some means “at least one”. The
representing the sets must overlap.

03/10/25Section 2.3 16

17. Example 2
Determining sets from a Venn Diagram
Solutions:
a.U = { a, b, c, d, e, f, g }
b.B = {d, e }
c.{a, b, c }
d.{a, b, c, f, g }
e.{d}
•Use the Venn Diagram to determine:
a.U
b.B
c.The set of elements in A but not B
d.The set of elements in U that are not in
B
e.The set of elements in both A and B.
03/10/25Section 2.3 17

18. The empty set 1
•If a set has zero elements, it is called the empty (or null) set
•Written using the symbol 
•Thus,  = { }  VERY IMPORTANT
•If you get confused about the empty set in a problem, try replacing  by
{ }
•As the empty set is a set, it can be a element of other sets
•{ , 1, 2, 3, x } is a valid set
18

19. The empty set 1
•Note that  ≠ {  }
•The first is a set of zero elements
•The second is a set of 1 element (that one element being the empty set)
•Replace  by { }, and you get: { } ≠ { { } }
•It’s easier to see that they are not equal that way
19

20. Set equality
•Two sets are equal if they have the same elements
•{1, 2, 3, 4, 5} = {5, 4, 3, 2, 1}
•Remember that order does not matter!
•{1, 2, 3, 2, 4, 3, 2, 1} = {4, 3, 2, 1}
•Remember that duplicate elements do not matter!
•Two sets are not equal if they do not have the same
elements
•{1, 2, 3, 4, 5} ≠ {1, 2, 3, 4}
20

21. Subsets 1
•If all the elements of a set S are also elements of a set
T, then S is a subset of T
•For example, if S = {2, 4, 6} and T = {1, 2, 3, 4, 5, 6, 7},
then S is a subset of T
•This is specified by S  T
•Or by {2, 4, 6}  {1, 2, 3, 4, 5, 6, 7}
•If S is not a subset of T, it is written as such:
S  T
•For example, {1, 2, 8}  {1, 2, 3, 4, 5, 6, 7}
21

22. Set operations: Union 2
22
U
A B
A U B

23. Set operations: Union 3
•Formal definition for the union of two sets:
A U B = { x | x  A or x  B }
•Further examples
•{1, 2, 3} U {3, 4, 5} = {1, 2, 3, 4, 5}
•{New York, Washington} U {3, 4} = {New York,
Washington, 3, 4}
•{1, 2} U  = {1, 2}
23

24. Set operations: Union 4
•Properties of the union operation
•A U  = A Identity law
•A U U = U Domination law
•A U A = A Idempotent law
•A U B = B U A Commutative law
•A U (B U C) = (A U B) U CAssociative law
24

25. Set operations: Intersection 2
25
U
BA
A ∩ B

26. Set operations: Intersection 3
•Formal definition for the intersection of two sets: A
∩ B = { x | x  A and x  B }
•Further examples
•{1, 2, 3} ∩ {3, 4, 5} = {3}
•{New York, Washington} ∩ {3, 4} = 
•No elements in common
•{1, 2} ∩  = 
•Any set intersection with the empty set yields the empty set
26

27. Set operations: Intersection 4
•Properties of the intersection operation
•A ∩ U = A Identity law
•A ∩  =  Domination law
•A ∩ A = A Idempotent law
•A ∩ B = B ∩ A Commutative law
•A ∩ (B ∩ C) = (A ∩ B) ∩ CAssociative law
27