Functions

Let A and B be two non-empty sets. Then a function ’f’ from set A to set B is a rule or method or correspondence which associates elements of set A to elements of set B such that

    i)  All elements of set A are associated to elements in set B.

    ii) An element of set A is associated to a unique element in set B.

In other words, a function ’f’  from a set A to a set B associates each element of set A to a unique element of set B.

If an element a є A is associated to an element b є B, then b is called the ‘f- image of a’ or ‘image of a under f’ or ‘the value of the function f at a’.  Also, a is called the pre-image of b under the function f we write it as b=f(a).

Illustration:

Let A = {1, 2, 3, 4} and B = {a, b, c, d, e} be two sets and let f₁, f₂, f₃ and f₄ be rules associating elements (A to elements of) B as shown in the following figures:

We observe that f₁ is not a function from set A to set B, since there is an element 3 є A which is not associated to any element of B. Also, f₂  is not a function from A to B because an element 4 є  A is associated to two elements c  and e  in B. But, f₃ and f₄ are functions from A to B. because under f₃ and f₄ each element in A is associated to a unique element in B.

Kinds of Functions:

One-One Function (Injection): A function  is said to be a one-one function or an injection if different elements of A have different images in .

Thus,  is one-one.

ALGORITHM:

STEP 1:

Take two arbitrary elements x, y (say) in the domain of f.

STEP 2:

Put  f(x) = f(y)

STEP 3:

Solve f(x) = f(y)

If f(x) = f(y) gives x=y only, then  is a one-one function (or an injection). Otherwise it is not.

Number of one-one functions:

If A and B are finite sets having m and n elements respectively, then

Number of one-one functions from A to B is the number of arrangements of n items by taking m at a time i.e., ⁿP_m.

Thus,

Number of one-one functions from A to B

Many – One Function:

A function  is said to be a many-one function if two or more elements of set A have the same image in B.

Thus,  is a many – one function if there exists x,y є A  such that x ≠ y  but f(x) = f(y)

Onto Function (Surjection):

A function  is said to be an onto function or a surjection if every element of B is the f- image of some element of A i.e., if f(A) = B  or range of f  is the co-domain of f.

Thus,  is a surjection if for each b є B Ǝ a є A such that f(a)=b.