Let A = { 10, 11, 12, 13, 14 }; B = { 0, 1, 2, 3, 5 } and fi : A → B , i = 1,2,3.

State the type of function for the following (give reason):

(i) f₁ = { (10, 1), (11, 2), (12, 3), (13, 5), (14, 3) }

(ii) f₂ = { (10, 1), (11, 1), (12, 1), (13, 1), (14, 1) }

(iii) f₃ = { (10, 0), (11, 1), (12, 2), (13, 3), (14, 5) }

There are several kind of function such as,

One-one function:

If f : A → B is a function then its treated as an one to one function if no element of B is associated with more than one element of A . A one-one function is also called an injective function.

Onto function:

A function f:A→B is called as an onto function if every element in B has a pre-image in A.

One-one and onto function:

A function f:A→B is called as an One-one and onto function if f maps all distinct elements of A with all distinct images in B and all element in B is an image of some element in A.

Constant function:

A function f:A→B is called as Constant function if all distinct elements of A has a single image in B.

Here we are given

A = { 10, 11, 12, 13, 14 }

B = { 0, 1, 2, 3, 5 }

fi:A → B , where i = 1,2,3.

(i) Here both 12 and 14 have same image i.e. 3 and there is no preimage of 0 of B in A, So it’s clear that the function given is neither an one to one nor onto.

(ii) Here it’s clear that for all element of A there is only one image in B i.e. 1.we can also write that for f₂:A→ B, f₂(x) = 1, ∀ x ∈ 1. So the above function is a constant function.

(iii) Here range of f₃ = {0,1,2,3,4,5} = B

It’s an one-one and onto function as all element in A have a distinct and non-repetitive image in B