One to One Function Proof

Introduction to one to one function  proof:

A function from A to B is called one to one if f(a) = f(b) then a = b. Each element in the function B has paired with only one  element in the function B. One to one function is a function in which all the variables in the domain has an individual range. Each element in the domain has the correspondent element in the range. Here we will see about one to one function proof.

One to One Function Proof:

A function is known as one to one if every element in the range of the function corresponds with one and only element in the domain.

Proof- one to one function proof

Let us consider the function, f : N -> N, it is defined as f(n) = n2  where N indicates the positive integers.

Assume an integer a and b in which f(a) = f(b).

The function is f(n) = n2, substituting a and b,

we have a 2 = b 2  ,  where  a 2 - b 2 = 0.

a 2 - b 2 = 0  - >   (a + b) (a - b) = 0

where  a + b  = 0 ,  a - b = 0

hence we have  a + b = 0   - >  a =  - b

hence we prove for a - b = 0   - >  a = b

the function f(a)= f(b)

Hence proved.

Example Problem - One to One Function Proof

Example problems 1 - One to one function  proof

F{(1, 0), (2, 3) ,( 4, 5), (6, 7)} show that the function is one to one function.

Solution:

Domain of the given function is 1, 2, 4, 6.

Range of the given function is 0, 3, 5, 7.

Hence the each of the domain has corresponding range individually.

Hence the given set exhibits one to one function.

Example problems 2 - One to one function  proof

Which of the following functions is one to one function?

F1{(1, 8), (2, 9) ,( 4, 10), (6, 11)}

F2{(1, 0), (2, 3) ,( 4, 8), (6, 9)}

F3{(1, 0), (2, 0) ,( 4, 5), (6, 7)}

Solution:

F1{(1, 8), (2, 9) ,( 4, 10), (6, 11)}

Domain of the given function1 is 1, 2, 4, 6.

Range of the given function1 is 8, 9, 10, 11.

Each element in the domain corresponds with the values of range.Understanding composite function is always challenging for me but thanks to all math help websites to help me out.

Hence the given function 1 is one to one function.

F2{(1, 0), (2, 3) ,( 4, 8), (6, 9)}

Domain of the given function2 is 1, 2, 4, 6.

Range of the given function 2 is 0, 3, 8, 9.

Each element in the domain corresponds with the values of range.

Hence the given function 2 is one to one function.

F3{(1, 0), (2, 0) ,( 4, 5), (6, 7)}

Domain of the given function 3 is 1, 2, 4, 6.

Range of the given function 3 is 0, 0, 5, 7.

The element in the domain 1 and 2 has the same range.

Hence the given function 3 is not one to one function.

Answer: F3{(1, 0), (2, 0) ,( 4, 5), (6, 7)} is not an one to one function.