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# Lecture - 4 Relations and Functions

**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.