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Functions and the types of functions

In this article, we will learn about the functions and the types of functions and also the types of mapping techniques in discrete mathematics.
Submitted by Prerana Jain, on August 15, 2018

Function

Suppose, X and Y are two any sets. A relation f from X to Y is said to be a function. If for every x E X there is a unique y E Y such that (x, y) E f. A function is a special case of the relation. The term such as "transformation", "mapping", "correspondence" and "operations" are used as synonyms for "function". The notation f: X → Y.

X → Y is used to express f as a function from X to Y. For a function f: X → Y if (x,y) E f then x is called an argument and the corresponding y is called the image of x under f. Instead of writing (x, y) e f, it is customary to write y= f(x) and to call y the values of the function f at x.

Example (f) : A -> B

function A, B
  • A is called domain of f.
  • B is called codomain of f.
  • The range is the set containing all images of the elements of an under function f. It is denoted by f(a).
  • Range of f = { f(n) | x E A}
  • Range of f C= B codomain

Types of functions

1. Constant function

The function f defined in a set X such that f(x) = a, xEX, is called a constant function. In other word f: X → Y is a constant function if the range of f consists of only one element. This can be represented by a diagram.

2. One-to-one (injective)

A mapping f of X into Y is said to be injective or one-to-one mapping. If distinct elements of X have distinct images in Y. It is called injective.

The f: X → Y is a (one-to-one) mapping, if and only if:f(x1) = f(x2) => x1 = x2

In other words f: X → Y is one-to-one (or injective) mapping, whenever x1 = x2 then,

F(x1) not equals to f(x2), where, x1, x2 belongs to X.

Thus a mapping from a set X into a set Y is one-to-one or injective, if each element of Y has at least one element of X mapping into Y.

3. Into mapping

A mapping f: A → B is said to be into mapping, if f(A) is a proper subset of B. In this case, we say that f maps A into B.

4. Onto function (surjective)

If the mapping f: X → Y is such that every element of Y is the image of at least one element of X then the mapping is called an onto or surjective mapping. In other words, the mapping f: X → Y is onto, if given yEY there exists an element xEX such that y = f(x).

5. One-to-one (Bijective) function

A mapping which is one-to-one as well as onto is called Bijective or one-to-one onto mapping.

To determine whether a mapping is Bijective, we follow the following procedure.

  • To show that f is one-to-one we must show that
    f(x1) = f(x2) => x1 = x2
  • To show that f is onto we must show that for each yEY, there exists an xEX such that f(x) = y.
    Then, it is one-to-one onto mapping. The sets X and Y have the same number of elements.

6. Invertible function

A function f: X → Y is said to be invertible. If there exists a function g: y → X such that: Fog = ty and gof = tx

Where Ix and Iy are the identify maps. In such a case the function g is called the inverse of f and is denoted by f^-1.






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