Function and Its Types.

In detail and In very simple method That can any one understand.
If you read this all you doubts about function will be clear.
because i have used very simple example and simple English words that you can pick quickly concept about functions.
#inshallah.

1. Mathematics.
Chapter: Function and Its types.
Awais Bakshy

2. Mathematics.
Chapter: Functions and its types.
Order pair.
Let a and b any two elements (a, b) is called order pair.
Abscissa or Domain.
Abscissa is a Latin word which mean spinal cord.
We take abscissa in English as domain.
The first element of order pair (a, b) is called abscissa or
domain.
Ordinate or Range.
The second element of the order pair (a, b) is called ordinate
or range.
Cartesian product.
Let “A” and “B” be any two non-empty sets. Then the set of
all those elements of the form (a, b) where �∈� and �∈�
is called Cartesian product of A and B.
It is denoted by A × B.
For Example.
A= {1, 2, 3} B= {a, b, c}
A × B= {1, 2, 3} x {a, b, c}

3. A × B= {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c), (3, a), (3, b),
(3, c)}
Key Points.
1. In general A × B ≠ B × A
2. If A × B=∅ then either A=∅ or B=∅
Binary Relation.
Let A and B are two non-empty sets then every subset of A ×
B is called a binary relation from set “A” to set “B”.
It is denoted by R. i.e. R ⊆ A × B.
Formula for Finding number of binary relation.
If we denote number of elements in set A is m and in set B is
n then the number of binary relation in A × B shall be??????
� �
.
For example.
A= {1} B= {x, y}
Then A × B= {(1, x), (1, y)}
Formula for binary relation.
m= 1 and n= 2
2
� �

2
1 ×2

2
2
=2
In A × B the numbers of binary relation are 2.

4. Domain of relation.
The set of all first elements of all order pair in a relation is
called domain of a relation.
It is denoted by Dom R
For Example.
R= {(1, a), (2, a) (3, c)}
Dom R= {1, 2, 3}
Range of relation.
The set of the second element of all ordered pair in relation
is called range of relation.
It is denoted by Ran R.
For Example.
R= {(1, a), (2, a) (3, c)}
Range R= {a, c}
Function.
Let A and B be two non-empty sets such that:
i. F is a relation A to B that is, f is a subset of A ×B.
ii. Domain of F= A
iii. Frist elements of any ordered pair in f is should not be
repeated.

5. Then f is called function from A to B. it can be written as F:
A→B.

For example.
R= {(1, a), (2, a), (3, b)}
Solution.
R= {(1, a), (2, a), (3, b)}
i. Dom R= {1, 2, 3}
ii. There is no repetition of first elements in ordered pair.
This relation follows both conditions so it is a function.
Domain of Function.
The set of the first element of the ordered pair of relation is
called its domain.
It is denoted by Dom (F).
For Example.
R= {(1, a), (2, a) (3, c)}
Dom F = {1, 2, 3}
Range of function.
The set of the second element of all ordered pair in relation
is called range of function.

6. It is denoted by Range (F)
For Example.
R= {(1, a), (2, a) (3, c)}
Ran F = {a, c}
Types of Function.
Into Function.
A function is said to be into function if Ran F ⊆ B.
For Example.
A= {1, 2, 3} B= {a, b, c}
Solution.
A × B= {1, 2, 3} x {a, b, c}
A × B= {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c), (3, a), (3, b),
(3, c)}
R = {(1, a), (2, a), (3, b)}
F = {(1, a), (2, a), (3, b)}
Ran F = {a, b}
Ran F ⊆ B
One to one Function.

7. A function is said to be one to one function if there is no
repetition in the second ordered pair in the function.
Or
A function F : A→B is called one to one function if all distinct
elements of set A has distinct image in set B.

For Example.
A= {1, 2, 3} B= {a, b, c}
Solution.
A × B= {1, 2, 3} x {a, b, c}
A × B= {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c), (3, a), (3, b),
(3, c)}
R = {(1, a), (2, b), (3, c)}
F = {(1, a), (2, b), (3, c)}
The above function is one to one function because there is no
repetition in second ordered pair.
Into and one to one (injective) function.
A function F : A→B is called injective function, if and only if
the function is into and one to one function.
For Example.
A= {1, 2, 3} B= {3, 4, 5, 6}

8. Solution.
A × B= {1, 2, 3} × {3, 4, 5, 6}
A × B= {(1, 3), (1, 4), (1, 5), (1, 6), (2, 3), (2, 4), (2, 5), (2, 6),
(3, 3), (3, 4), (3, 5), (3, 6)}
R= {(1, 4), (2, 5), (3, 6)}
F= {(1, 4), (2, 5), (3, 6)} this function is into and one to one
function so this function is called injective function.
Onto Function (Surjective Function).
A function is said to be onto function if Ran F= B.
For Example.
A= {1, 2, 3} B= {a, b, c}
Solution.
A × B= {1, 2, 3} x {a, b, c}
A × B= {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c), (3, a), (3, b),
(3, c)}
R= {(1, a), (2, b), (3, c)}
F= {(1, a), (2, b), (3, c)}
Ran F= {a, b, c} is a Surjective function.
One to one and onto (Bijective) Function.

9. A function F : A→B is called bijective function, if and only if
the function F is onto and one to one function.
For Example.
A= {1, 2, 3} B= {a, b, c}
A × B= {1, 2, 3} x {a, b, c}
A × B= {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c), (3, a), (3, b),
(3, c)}
R= {(1, a), (2, b), (3, c)}
F= {(1, a), (2, b), (3, c)}
Ran F= {a, b, c}
There is no repetition in the second ordered pair.
Above function is a bijective function.
Exercise.
Q No 1.
If A= {6, 5, 3} B= {1, 2} then find two binary relation of A×B
and also find there domain.
Q No 2.
If A= {2, -1, 3} then write three binary relation for A×A also
find the domain and range of these binary relation.
Q no 3.

10. Write the number of the following.
1. In A×B the number of element in set A is 5 and in set B
is 3.
2. The number of elements in set A is 2 and in set B is 4.
Q No 4.
If A= {1, 2, 4} B= {1, 3, 5. 7} then write the binary relation of
A×B when R= {(x, y)/x∈A∧y∈B∧y<x}
Q No 5.
If A= {1, 2, 3} B= {2, 3, 4} then write binary relation in A×A
and A×B when R= {(x, y) x∈A∧y∈B∧y>x}

Q No 6.
If A= {2, 0, 2} B= {-1, 0, -2} then write binary relation for all
in A×B if R= {(x, y) x∈A∧y∈B∧y≤x}
Q No 7.
Set of Whole Number Then Find.
R= {(x, y) x, y∈W∧x+y=7}
Q No 8.
If A= {2, 4, 8} B= {0, 3, 5} then find the function of the
following relation also find (Into) (Onto) (one to one) and
bijective function.
R1= {(2, 0), (4, 3), (2, 5)}
R2= {(4, 0), (3, 3), (8, 0)}

11. R3= {(8, 3), (4, 5), (2, 0)}