The document defines and provides examples of key concepts related to sets: 1) A set is a collection of distinct objects that can be defined using a description, set notation, or a Venn diagram. 2) Sets have elements and can be equal, subsets, or complements of one another. 3) Common set operations include intersection, union, and complement which combine elements of sets in different ways. 4) Venn diagrams can visually represent relationships between sets and their operations.

1. SET
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2. SETS
 A group of objects
 A collection of distinct objects, considered as
an object in its own right.
 Sets are one of the most fundamental
concepts in mathematics
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3. SETS
Example:
i. Set of even numbers less than 10
2,4,6,8
ii. Set of vowels : a,e ,i,o,u
iii. Colours of rainbow : red, orange, yellow,
green, blue, indigo and violet
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4. SETS
Can be defined by
* description
* using set notation,
e.g i. A set of students of WMS
ii. 2,3,5,7
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5. Sets are named with capital letters and the
objects are enclosed in braces
e.g : Set A is a set of the first seven numbers in
multiple of 7 ( description)
A = 7,14,21,28,35,42,49
(set notation)
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6.  The objects are known as elements of the
set.
set notation /symbol :
: is not an element of
A = 7,14,21,28,35,42,49
21 is an element of set A
21 A
8 A
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7. Set also can be represented by a Venn diagram
Venn Diagram: an enclosed geometrical figure,
i.e circle, triangle, rectangle
e.g P = factors of 9 = 1,3,9
P
.3
.9 .1
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8. n ( A ) means the number of elements of set A.
e.g i. P = 1,4,9,16 ,25
n(P)=5
ii. Q = AVATAR
Q = A, V, T ,R
n (Q) = 4
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9. P
.3 P= 3
P
3 n( P ) = 3
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10. Empty Sets : A set which does not have any
elements
Symbol : or
e.g Set of odd numbers which can be divided by
2.
=
e.g A = A polygon with two sides
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11. Equal sets:
Consider sets A= 1,2,3,4 and B = 1,4,2,3 . All
elements of set A are elements of set B , hence set A
is equal to set B.
A = B
e.g K = 11,12,13,14 and L = 11,14,12
The element 13 is not found in set L, therefore, K
and L are not equal sets.
K L
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12. e.g Given that
A = prime numbers less than 15
B = 2, 3, 5, 11, 13 , y +2
and A = B, find the value of y.
Solution :
A = 2, 3, 5, 7, 11, 13
y+2=7
y=5
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13. SUBSET
Set P is a subset of set Q if all the elements of
set P are also found in set Q.
“ Set P is subset of set Q”
and is written as
P Q
“ set P is not a subset of set Q” is denoted by
P Q
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14. e.g Given that
A = even numbers from 2 to 16
B = 4, 16
C = perfect squares less than 20 . State
whether each of the following is true or false.
a) B A b) B C c) C A d) A B
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15. Venn Diagram : P Q
Q
P
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16. P = 5, 6, 10 and Q = 6,10 .
P Q
.10
.5 .6
Q P
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17. Listing the subsets of a set .
i. P = 2
Subsets of P : , 2
ii. P = 2, 4
Subsets of P : , 2 4 2,4
iii. P = 2,4,6
Subsets of P = , 2 4 , 6 2,4 2,6
4,6 , 2,4,6
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18. # 1 : an empty set is a subset of any set
#2 : A set is a subset of itself
#3 : number of subsets of set P is given by
n ( A)
2
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19. Universal set.
- the set that contains all the elements under
discussion and is denoted by the symbol
.
- any set under discussion is a subset of
the universal set.
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20. 20

21. e.g Given that = x: 4 x 10, x is an
integer
P = x: x is a multiple of 3
a. List all the elements of and P.
b. Draw a Venn diagram to represent and
P.
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22. Complement set :
The complement , A’ , of set A is a set which
contains all elements in the universal set, ,
which are not elements of set A.
e.g = 2,3,4,5,6,7,8
A = 3,5,7,8
A’ = 2, 4,6,
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23. A
A’
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24. OPERATIONS ON SETS
Intersection of two sets.
The intersection of
two sets , A and B, is
the set of all the
common elements of A
and B, and is denoted
as
A B
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25. Intersection of two sets
The intersection of two sets , A and
B, is the set of all the common
elements of A and B, and is
denoted as
A B
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26. A B
A B
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27. e.g Given P = 3, 4, 5, 7 , Q= 4,5, 7, 8 and
R = 6, 8 , 9 . Find the following set;
a. P Q b. Q R c. P R
Solution:
a. 4,5,7
b. 8
c. or
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28. Relationship between A B and A or B.
( A B ) A and ( A B) B
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29. P Q
R
Shade the region that represents
P Q R’
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30. e.g In a class of 40 students, 22 of them play
badminton while 24 play football.Among
these students,14 students play both
badminton and football.
a. Draw a Venn diagram to represent the
information above
b. Find the number of students who play
neither badminton nor football.
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31. Union of sets.
- The union of two sets , A and B is a set that
consists all the elements found in the sets A
and B.
- Denoted by A B ( A union B)
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32. e.g A = 2,4,6,8 and B = 4,8,10
A B = 2,4,6,8,10
n( A B )= 5
- A (A B ) and B (A B)
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33. A computer centre has 120 students. 76 of
them study Java while 60 of them study
Pascal. Given that 35 of them study both.
Find the number of students who study Java
or Pascal.
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