This document covers the topic of functions and their types in mathematics, explaining concepts such as ordered pairs, Cartesian products, binary relations, and the definition of functions. It describes key terms including domain, range, and various types of functions such as injective, surjective, and bijective, along with examples and formulas. Additionally, exercises are provided to reinforce understanding of these concepts.

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

2. Mathematics.
Chapter:Functionsand its types.
Orderpair.
Let a and b any two elements (a, b)is calledorder pair.
Abscissaor 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.
OrdinateorRange.
The second element of the order pair (a, b)is called ordinate
or range.
Cartesianproduct.
Let “A” and“B” be any two non-empty sets. Then the set of
all those elements of the form (a, b)where 𝒂 ∈ 𝑨 and 𝒃 ∈ 𝑩
is called Cartesianproduct 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=∅
BinaryRelation.
Let A and B are two non-empty sets then every subset of A ×
B is called a binary relationfrom 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 relationin A × B shall be𝟐 𝒎 ×𝒏
.
For example.
A= {1} B= {x, y}
Then A × B= {(1, x), (1, y)}
Formula for binary relation.
m= 1 and n= 2
2 𝑚 ×𝑛
21 ×2
22
=2
In A × B the numbers of binary relationare 2.

4. Domainofrelation.
The set of all first elements of all order pair in a relationis
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}
Rangeof 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 calledfunction 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 relationfollows both conditions so it is a function.
DomainofFunction.
The set of the first element of the ordered pair of relationis
called its domain.
It is denoted by Dom (F).
For Example.
R= {(1, a), (2, a) (3, c)}
DomF = {1, 2, 3}
Rangeof 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}
Typesof Function.
IntoFunction.
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
Oneto oneFunction.

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 iscalledone 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.
Intoandonetoone(injective)function.
A function F:A→Bis calledinjective 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 calledinjective function.
OntoFunction(SurjectiveFunction).
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.
Oneto oneandonto(Bijective)Function.

9. A function F:A→Bis calledbijective function, if and only if
the function Fis 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 bijectivefunction.
Exercise.
Q No 1.
If A= {6, 5, 3} B= {1, 2} then find two binary relationof A×B
and also find there domain.
Q No 2.
If A= {2, -1, 3} then write three binary relationfor 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 relationof
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 relationin 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 relationfor 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)}