0

Guest

Courses New

Using the definition, prove that the function f : A→ B is invertible if and only if f is both oneone and onto.

 Using the definition, prove that the function f : A→ B is invertible if and only if f is both oneone and onto.

Grade:12

1 Answers

Harshit Singh
askIITians Faculty 5963 Points
3 years ago
Dear Student

Let us assume
f: A → B be many-one function.
Let’s assume
f(a) = p and f(b) = p
So, for inverse function we will have f^-1(p) = a and f^-1(p) = b
Thus, in this case inverse function is not defined as we have two images ‘a and b’ for one pre-image ‘p’.
But for f to be invertible it must be one-one.
Now, let f: A → B is not onto function.
Let B = {p, q, r} and range of f be {p, q}.
Here image ‘r’ has not any pre-image, which will have no image in set A.
And for f to be invertible it must be onto.
Hence, ‘f’ is invertible if and only if ‘f’ is both one-one and onto.

A function f = X → Yis invertible iff f is a bijective function.

Thanks

Think You Can Provide A Better Answer ?

Other Related Questions on Magical Mathematics[Interesting Approach]

View all Questions »

ASK QUESTION

Get your questions answered by the expert for free

Answer and earn gift vouchers

Win Gift vouchers upto Rs 500/-

View courses by askIITians

Design classes One-on-One in your own way with Top IITians/Medical Professionals

Click Here Know More

Complete Self Study Package designed by Industry Leading Experts

Click Here Know More

Live 1-1 coding classes to unleash the Creator in your Child

Click Here Know More

a Complete All-in-One Study package Fully Loaded inside a Tablet!

Click Here Know More