Back to: O level mathematics notes syllabus uganda

**Definition:**

A set is a collection of objects or members, which are related in some

way. Sets are denoted by capital letters, e.g. set A, B, M etc.

Terms used:

**Member (element) of a set**

The objects in a set are called members or elements of the set.

Members of a set are enclosed in curly brackets. The symbol Є is a

short form of saying ‘’is a member of’’ and is for ‘’not a member

of’’. Consider set A = {1, 3, 4, 6}. Then 1ЄA, 3ЄA, 4ЄA, and 6Є A.

Whereas 7, 8, 9 etc are not members of set A, i.e. 7 A, 8 A, 9 A,

etc.**Subsets**

Set A is said to be a subset of set B if every element of set A is also in

set B. E.g. given that B = {1, 2, 3, 4, 5, 6, 7, 8} and A = {1, 3, 5, 7}. Here

every element of set A is also in set B. Therefore, set A is a subset of

B. The symbol < or > is for ‘’subset of’’ and is for ‘’not a subset

of’’. Therefore A < B or B > Abut B A because not all elements of

B are in A.**Empty set (null set)**

An empty set is a set with no element. It is at time called null set. The

null set is denoted by the symbol { } or ф.

Note:

The empty set { } is not the same as {0}. This is because the set {0} has

one element which is 0 whereas the set { } has no element.**Finite sets**

The set is called finite if the elements of the set can be counted.

Example

Consider the following sets:

D = {days of the week}

F = {factors of 12}

G = {whole numbers greater than 5 but less than 11}

We can list all the members of these sets.

D = {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday,

and Sunday}

F = {1, 2, 3, 4, 6, and 12}

G = {6, 7, 8, 9, and 10}

**Infinite sets**

These are sets with unlimited number of elements.

Example

Given the following sets:

W = {whole numbers}

R = {real numbers}

M = {multiple of 3}

Here, we cannot list all the members of these sets.

W = {0, 1, 2, 3, 4, 5, 6 ……}

R = {……-2, -1, 0, 1, 2 …}

M = {3, 6, 9, 12 ……}

All members of these sets cannot be exhausted so they are infinite

sets.**Number of elements in a set**

The number of element in a finite set can be counted. The number of

elements of set A is denoted by n A and it is the total number of

elements in set A.**Example**

Find the number of elements in the following sets.

a) R = {1, 2, 3, 4, 5, 6, 7, 8, 12}

b) B = {2, 4, 6, 8, 9}

Solution

a) n (A) = 9

b) n (A) = 5**Example**

Given that set B = {factors of 24}

a) Write out set B in full

b) Find n (B)

Solution

a) B = {1, 2, 3, 4, 6, 8, 12, 24}

b) n (B) = 8**Example**

Given that set N = {natural numbers from 2 to 11}

a) Write out set N in full

b) Find n (N)

Solution

a) N = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11}

b) n (N) = 10

**Equal sets**

Two or more sets are equal if they contain the same elements.

E.g. A = [1, 3, 5, 7} and B = {1, 3, 5, 7} are equal sets. Here A = B and

also B = A**Equivalent sets**

Two or more sets are said to be equivalent if they contain the same

number of elements. E.g. set A = {a, e, i, o, u} and B = {2, 4, 6, 8, 10}.

Sets A and B contain the same number of elements which is 5. We

therefore say that they are equivalent sets.**Union of sets ( u )**

The union of two sets is the set of all elements that are members of

either set. The symbol for union is U.

Example

Given that: M = {1, 2, 3, 4} and N = [3, 4, 6, 7}

i) List M U N

ii) Find n M U N**Solution**

i) M U N = (1,2,3,4,6,7)

ii) n (M U N) = 6**Intersection of sets ( n )**

The intersection of two sets or more sets is the set of elements that

are in both sets.**Example**

Given two sets: A = {–1, 0, 4, 5, 6, 7} and B = {–1, 6, 8, 10}

Find:

i) n( A u B)

ii) n(A n B)**Solution**

i) (A u B) = { -1,0,4,5,6,7,8,10} n(A u B) = 8

ii) (A n B) = {1,6} .: n (A n B) = 2**Disjoint set**

When the intersection of the two sets is empty, the two sets are

called disjoint sets. E.g. given that P = {1, 3, 5, 7} and Q = {2, 4, 6, 8}.

Here( P n Q) = { }**Complement of set**

Consider two sets: A = {a, b, c, d} and B = {a, b, c, d, e, f}.

Members which are present in B and not present in A is called

complement of A denoted by Á or Â. From the two sets above:

A’ = { e,f } therefor n(A’ ) = 2

Also:

A’ n B = {e ,f } = n (A’ n B ) = 2**The universal set**(ℇ)

This is a set that contains all the members of an item or object under

consideration. It is denoted by the symbol ℇ.