Relations And Functions
Relations and Functions

Relation: A relation from a set A to a set B is a subset of the Cartesian product A × B.
 Think of a relation as a mapping between two sets.
 The elements of the relation are ordered pairs.

Domain and range: The domain of a relation R from A to B is the set of all first components of ordered pairs in R. The range of R is the set of all second components of ordered pairs in R.
 The domain tells you which set the relation starts with, and the range tells you which set it ends with.
 The Cartesian product A × B is the set of all ordered pairs (a, b) where a is in A and b is in B.

Function: A function from A to B is a relation from A to B that assigns to each element of A exactly one element of B.
 A function is a special type of relation where each input corresponds to a single output.
 Functions are also called mappings.

Injective function: A function f from A to B is injective (or onetoone) if, for all a1, a2 ∈ A, f(a1) = f(a2) implies a1 = a2.
 An injective function preserves distinct elements.
 In other words, if the outputs of an injective function are equal, then the inputs must also be equal.

Surjective function: A function f from A to B is surjective (or onto) if, for every b ∈ B, there exists an a ∈ A such that f(a) = b.
 A surjective function covers the entire range.
 In other words, every element in the range is the output of at least one input in the domain.

Bijective function: A function f from A to B is bijective if it is both injective and surjective.
 A bijective function is both onetoone and onto.
 Bijective functions are also called invertible.

Inverse function: If f is a bijective function from A to B, then there exists a unique function g from B to A such that f(g(b)) = b and g(f(a)) = a for all a ∈ A and b ∈ B. The function g is called the inverse of f and is denoted by f^1.
 The inverse of a function undoes the original function.
 Inverse functions only exist for bijective functions.

Composition of functions: If f is a function from A to B and g is a function from B to C, then the composition of f and g, denoted by g o f, is the function from A to C defined by (g o f)(a) = g(f(a)) for all a ∈ A.
 Composition of functions allows you to combine multiple functions to create new functions.
 The order of functions in composition matters.