**WHAT
IS A SET IN MATHEMATICS**

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A set is a collection
of distinct objects. These objects must be well defined
and must have something in common. It could be a group
of numbers, persons etc. For instance, the students
of a particular class or taking a particular subject
or writing a particular exam can form a set. Also,
a list of even numbers between 0 and 50 can form a
set. We could have a set of integers between 1 and
20.

There are two ways of writing a set: (a) Enumeration,
and (b) Description.

(a) Enumeration method: When we use enumeration to
write a set, we will write out all the individual
elements of the set. For instance, if set T is all
real numbers from 1 to 5, then, we can write out the
elements of the set thus: T = {1, 2, 3, 4, 5}.

(b) Description method: This is the use of statement
or symbol to describe the elements of a set. For instance,
if set S is all positive integers, we will write the
set as: S = {x | x a positive integer} which reads:
"S is the set of all x-numbers, such that x is
a positive integer". Note that a vertical bar
(or use colon) is inserted to separate the general
symbol for the elements from the description of the
elements. Using another example, if set M is all real
numbers greater than 2 but less than 6; we can express
this symbolically thus: M = {x : 2 < x < 6}.

The objects
in a set are called the elements of the set and they
are indicated by the symbol ,
which is read as: "is an element of". So
if a set A is defined as: A = {2, 4,6, 8}. The elements
of the set are 2, 4, 6, 8. Therefore 2
A, meaning 2 is an element of A. Also, 4
A, 6
A, and 8
A.

However, if an object is not an element of a set,
it is indicated by the symbol ,
which is read as: "is not an element of".So
if a set A is defined as: A = {2, 4, 6, 8}. It is
obvious that 1, 3, 5 are not elements of the set.
Therefore 1
A, meaning 1 is not an element of A. Also, 3
A, and 5
A.

The number of elements in a set is called the cardinality
of the set and it is indicated by the symbol "n".
In the case of set A = {2, 4, 6, 8}. The set contains
4 elements, therefore the cardinality of A = 4 i.e
n{A} = 4. Using another example, if a set B = {1,
3, 5, 7, 9}; n{B} = 5. If set J = {2, 5, a, c, 7,
e, 8, f}; n{B} = 8.

A set that contains a finite or countable number of
elements is called finite set; for example A = {2,
4, 6, 8} contains four (4) elements which is countable.
A set that contains an infinite or uncountable number
of elements is called infinite set; for example S
= {x | x a positive integer} contains all positive
integers i.e 1, 2, 3, 4 etc which is uncountable.
A set that contains no element is called an empty
or null set and it is represented by the symbol or
.
Note that it is not the same as
because there is an element in the latter while
does not contain any element. So,
is not an empty set because it contains as
element.

**RELATIONSHIP
BETWEEN SETS**

There are many possible kinds of relationship that
can exist between two sets. Lets examine some of them:

**(A) Equal
Sets**: Two or more sets are said to be equal
if they have the same elements irrespective of their
arrangement. For example, if set A = {2, 4, g, h,
6, 8}, and set B = {6, h, 4, 8, g, 2}. Since the same
elements that are in set A are also in set B then,
A = B.

**(B) Subset
of a set**: A set is a subset of another set
if **all** its elements are contained
in the other set. For example, if set M = {3, 6, 1,
5, 4} and set N = {1, 3}. It is obvious that all the
elements of set N are in set M. So N is a subset of
M. The symbols
of subset are:
(is contained in) and
(includes); so we can write N
M (N is contained in M) or we write M
N (M includes N). Also note that the symbol
means not a subset of or not contained in, and the
symbol
means not include. If P = {2, 9, 8}, it is obvious
that the elements of set P are not in set M, so we
can write that P
M (P is not a subset of or not contained in M) or
M
P (M does not include P). Also, if Q = {7, 6, 5, 2},
you will observe that some elements of Q are contained
in M (e.g. 6, 4) but some elements are not (e.g. 7,
2). Since not all the elements of Q are in M then,
Q
M (Q is not a subset of or not contained in M).

It is important to
note that the subsets of a set include the set itself
and also an empty set as well as all the possible
combination of the elements. The total number of subsets
that can be obtained from a set can be known using
the formular
where n is the number of elements in the set. Lets
take some examples to illustrate this.

**Question**:
Given that A = {2, 5}, write out all the subsets of
A.

**Answer**:
The first thing to do is to know the total number
of subsets that can be obtained from set A by using
the formular .
Also, you must remember that the subsets of a set
includes the set itself and an empty set.

Total number of subset =
where n is the number of elements in the set.

Since Set A has 2 elements, then, the total number
of subsets for set A =
= 4.

The subset of A =
{2, 5} are: first {2}, second {5}, third {2, 5}, and
fourth { }.

**Question**:
Given that Z = {b, f, s}, write out all the subsets
of Z.

**Answer**:
The total number of subsets that can be obtained from
set Z can be known using the formular
where n is the number of elements in the set. Also,
the subsets of all sets include the set itself and
an empty set.

Since Set Z has 3 elements, then, the total number
of subsets for set Z =
= 8.

The subsets of set
Z are: {b}, {f}, {s}, {b, f}, {b, s}, {f, s}, {b,
f, s}, and { }.

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