Is the Square Root Function Bijective in all Branches of Mathematics?

In summary, the square root function is bijective if the domain is the non-negative real numbers and the codomain is the non-negative real numbers. To prove this, we must show that the function is both injective (one-to-one) and surjective (onto). To show injectivity, we must show that for any two inputs x and y, if the square root of x is equal to the square root of y, then x must equal y. This can be easily shown by taking the square of both sides of the equation. To show surjectivity, we must show that for any output y, there exists an input x such that the square root of x is equal to y. This can be shown by taking the square of
  • #1
aaaa202
1,169
2
In what branches of mathematics is this proven.. I have never seen a proof, so I wonder if anyone can give me the basics of what is done to proove it or got a link to a proof..

Edit: By square root I mean the positive square root.
 
Physics news on Phys.org
  • #2
to prove square root function to be bijective you need to specify the domain and codomain of the function.
if codomain is set of real numbers, then its not bijective.
 
  • #3
aaaa202 said:
In what branches of mathematics is this proven.. I have never seen a proof, so I wonder if anyone can give me the basics of what is done to proove it or got a link to a proof..
Is this a homework assignment? You should know by know that we won't do your assignments for you, but we'll guide you while you do the work.
aaaa202 said:
Edit: By square root I mean the positive square root.
 
  • #4
This is definitely not a homework assignment. I'm on holidays and this speculation came to me. I study physics and thus my courses do not focus on how to get to the mathematical results rigorously but rather their applications. Therefore I have to do this work on my own and I have no idea whatsoever what tools will be needed for the above proof. The domain of the positive square root are all positive integers including zero. But to go from here to show that the square root takes one number into another number uniquely I don't know how to show. Thus I ask you.
 
  • #5
Then why post it in "Homework and Classwork"? In any case "the square root takes one number into another number uniquely" is NOT "bijective". That is "injective". You can't say "bijective" without, as pcm said, specifying the domain and codomain.

If you intend the domain and codomain as "the non-negative real numbers" then, yes, the square root function is bijective. To show that you show it is "injective" ("one to one"): if [itex]\sqrt{x}= \sqrt{y}[/itex] then x= y. That's easy to show. And "surjective" ("onto") is equally easy- given any non-negative number, y, show that there exist a non-negative number x such that [itex]\sqrt{x}= y[/itex].
 
  • #6
aaaa202 said:
The domain of the positive square root are all positive integers including zero.
HallsOfIvy alluded to this, but to make it more explicit - the domain of this function is all nonegative real numbers, not just the nonnegative integers.
 

Related to Is the Square Root Function Bijective in all Branches of Mathematics?

1. What is the definition of bijective?

Bijective refers to a function or mapping between two sets where each element in the first set is paired with exactly one element in the second set, and vice versa. This means that there is a one-to-one correspondence between the two sets.

2. How is bijectivity related to square root?

In mathematics, a function is considered bijective if it is both injective (one-to-one) and surjective (onto). The square root function is an example of a bijective function because every positive number has a unique positive square root, and every non-negative number has a square root.

3. What is the proof of square root being bijective?

The proof of square root being bijective involves showing that the function is both injective and surjective. This can be done by using algebraic manipulation and properties of square roots, such as the fact that the square root of a product is equal to the product of the square roots.

4. Why is it important to prove that square root is bijective?

Proving that square root is bijective is important because it confirms that the function is well-defined and has a unique inverse. This allows for the use of the inverse function to solve equations and make other mathematical calculations.

5. Are there any exceptions to the bijectivity of square root?

Yes, there are exceptions to the bijectivity of square root. For example, the square root function in the complex numbers is not bijective because each complex number has two square roots. Additionally, the square root function is not bijective for negative numbers when considering only real numbers, as there is no real number that can be squared to equal a negative number.

Similar threads

Quadratic equation and its roots
    • Precalculus Mathematics Homework Help
Replies
14
Views
1K
I Reason for the irrationality of (most) square roots?
    • General Math
Replies
12
Views
2K
Minimum of x+1/x (Perimeter Square < Perim equal area rects)
    • Precalculus Mathematics Homework Help
Replies
1
Views
2K
I I think I've hit upon a very easy proof of impossibility of quintic
    • Linear and Abstract Algebra
Replies
2
Views
990
To find the range of a given ##\sin## function
    • Precalculus Mathematics Homework Help
Replies
15
Views
712
Foundations Looking for Introduction to Proofs
    • Science and Math Textbooks
Replies
3
Views
1K
Root 2 irrationality proof (geometric)
    • Precalculus Mathematics Homework Help
Replies
1
Views
2K
Fast reciprocal square root algorithm
    • Programming and Computer Science
Replies
30
Views
4K
Dedekind Cut, as stated by Richard Dedekind
    • STEM Educators and Teaching
Replies
5
Views
2K
A Integrating a function of which poles appear on the branch cut
    • Calculus
Replies
1
Views
2K

Hot Threads

Recent Insights

Back
Top