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Functions in Set Theory

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Functions in Set Theory

functions-in-set-theory

Functions in Set Theory

Introduction:

The concept of a function is one of the most key ones in mathematics. The word "map," "mapping," "transformation," and many more all refer to the same thing; which one to use in a given context is typically chosen by tradition and the user's mathematical background. Gottfried Wilhelm Leibniz used the word "function" for the first time in a text he sent in 1673 to refer to a number associated to a curve's points, such as a coordinate or slope.:

Before defining the term function of a set, it is necessary to understand the terminology of image.

Image of a Function:

The collection of all possible output values is known as a image of a function in mathematics.

Definition of Function:

Assume that we assign a distinct element from a set X to each element of a set Y. This collection of mappings is known as a function from X into Y. Set X is referred to as the function's domain, while Set Y is referred to as the target set or codomain.

Mathematical Representation of Function:

In most cases, symbols are used to represent functions. Let f is a function from set X to set Y write,

f: X →Y

Example of Function:

Consider the function f (x) = x^2, i.e., f assigns to each real number its square. Then the image of 2 is 4, and so we may write f (2) = 4.

Functions as Relations:

A relation from X to Y (i.e., a subset of X × Y) is defined by the function f: X→ Y such that each x ∈ X is a member of a unique ordered pair (x, y) in f.

Although we do not distinguish between a function and its graph, we will still refer to f as a collection of ordered pairs as "graph of f". Additionally, as the graph of f is a relation, its picture can be drawn, just like relations in general, and this picture-based representation is sometimes referred to as the graph of f.

Additionally, the geometric requirement that each vertical line cross the graph at exactly one point is identical to the defining condition of a function, which states that each a X belongs to a certain pair (x, y) in f.

Composition of Functions:

Suppose that a functions f: X → Y and g: K → L; that is, where the codomain of f is the domain of g. Then we may define a new function from X to L, called the composition of f and g and written gof, as follows: (gof) (x) ≡ g (f (x))

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Types of Functions:

There are three types of functions.

  • One-to-One Function.
  • Onto Function.
  • Invertible Function.

One-to-One Function (Injective)

Definition:

A function f: X → Y is said to be one-to-one (1-1) if different elements in the domain X have distinct images. The same thing may also be expressed as that f is one-to-one if f (x) = f (x`) implies x = x`.

Example:

The function f(x) = x + 2 is a one-to-one function because it produces different output for a different input of x. Put x = 1

F(x) = 1+2

F(x) = 3

Put x = 7

F(x) = 7+2

F(x) = 9 and so on….

One-to-One Function

One-to-One Function

Onto Function (SURJECTIVE)

Definition:

A function f: X → Y is said to be onto function if each element of Y is the image of some element of X. In other words, f: X → Y is onto if the image of f is the entire codomain, i.e., if f (X) = Y. In such a case we say that f is a function from X onto Y or that f maps X onto Y.

Surjective Function

Surjective Function

Invertible Function

Definition:

A function f: X → Y is invertible if it’s inverse relation f ^-1 is a function from X to Y. In other words, if a function f satisfies the conditions of into as well as onto then the function f is said to be invertible function.

Theorem:

A function f: X → Y is invertible if and only if f is both one-to-one and onto.

Invertible Function

Invertible Function

Geometrical Characterization of One-to-One and Onto Functions:

Now think about functions with the notation f: R→ R because such graph of function can be plotted in the Cartesian plane R2 and because functions can be recognized by their graphs. if the concepts of being one-to-one and onto have any geometrical significance.

Accordingly,

(1) If each horizontal line crosses the graph of f at most once, then f:R →R is one-to-one.

(2) Each horizontal line must intersect the graph of f at one or more points in order for it to be an onto function.

Specifically, each horizontal line will cross the graph if f is both one-to-one and onto, or invertible at exactly one point.

This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.

© 2022 Kinza Javaid

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