जिसने कहा कल, दिन गया टल,
जिसने कहा परसों,बीत गए बरसो
जिसने कहा आज, उसने किया राज।

Logic for this lecture:

• Functions \( \subseteq \) Relations
• A relation from non-empty set A to non-empty set B is said to be function, if
→ every element of set A has only one image in set B.
In other words, Domain = A and
no distinct ordered pairs have same first element.

DIFFERENT MAPPINGS OF FUNCTIONS

One-One / Injective Mapping / Monomorphism

A function is said to be one-one if each member of the range of the function arises for one and only one member of the domain of the function.
• A function f: A → B is said to be one-one if  \( f(a) = f(b) \Rightarrow a = b \)

Many-One

A function is said to be many-one if at least one member of the range of the function arises for more than one member of the domain of the function.
• A function f: A → B is said to be one-one if  \( f(a) = f(b) \Rightarrow a \ne b \)

Onto Mapping / Surjective Function

• In a Surjective function,
Range = Co-domain

Into Mapping

• In a Surjective function,
\( Range \subset Co-domain \)

• A function which is both one-one and onto is called Bijective Function.