Definition: Let X and Y are two sets. A relation f from X to Y is a function if for every x ∈ X there is a unique element y ∈ Y, such that (x, y) ∈ X
Here we denote it as f: X → Y, which means f is a function from X to Y.
Hence, We represent it as-
Example: Let And And
Hence we represent the function as below
Here A is the domain of f and B is the co-domain of f
Surjective function (Onto-Mapping)
A mapping f: X → Y said to be onto-mapping if the range set .
If f: X → Y is onto then each element of B if f-image of at least one element of X.
A mapping is a into if it is not onto.
Injective fn (One-To-One Mapping)-
A mapping f: X → Y is said to be one-to-one mapping if distinct elements of X are mapped into distinct elements of Y.
That means if f is a one-to-one mapping, then-
Or in other words,
Bijective functions (One-To-One, Onto)-
A mapping f: X → Y is the bijective mapping if it is both one-to-one and onto.
Identity function (Identity Mapping)-
If f: X → Y is a function such that every element of X is mapped onto itself then f is said to be an identity mapping.
We denote identity mapping by
If is an identity mapping.
Let X and Y be sets and A be a subset of X. A function f from A to Y is called a partial func. from X to Y. The set A is a called the domain of f.
Let f: X → Y then f is a constant function if every element of X is map onto the same element of Y.
Let f: X → Y is a one-one and onto mapping, then is called the inverse mapping of f.
We define the Inverse mapping as
Composition of Functions
Let f: X → Y and g: Y → Z are two mappings, then we denote the composition of these two mappings f and g by gof is the mapping from X into Z.
Hence, we define it as
That means is a mapping defined as below-
A function f from a set X to a set Y is an invertible func if there exists a function g from Y to X, such that-
f(g(y)) = y And g(f(x)) = x For every x ∈ X and y∈ Y
Recursive Functions- A function f is a Primitive recursive if we obtain it from the initial functions by a finite number of operations of composition and recursion.
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