Injective Property of Rotation Function (x,y) to (y,x)

You can show that the function is injective by demonstrating that for any two inputs (x,y) and (x',y'), if the outputs (y,x) and (y',x') are equal, then the inputs must also be equal. This can be done by showing that y=y' and x=x', as you have correctly summarized. In summary, the function is injective because for any two inputs (x,y) and (x',y'), if the outputs (y,x) and (y',x') are equal, then the inputs must also be equal. This is demonstrated by showing that y=y' and x=x'.
  • #1
dpa
147
0
Hi all,

Q. A function takes (x,y) and gives (y,x). Is this function injective?

For any function to be injective, f(x,y)=f(x',y')=>(x,y)=(x',y').
But here, I get,
(y,x)=(y',x')
How can I show the function is injective? It appears to be one.

Thank You.
 
Physics news on Phys.org
  • #2
dpa said:
Hi all,

Q. A function takes (x,y) and gives (y,x). Is this function injective?

For any function to be injective, f(x,y)=f(x',y')=>(x,y)=(x',y').
But here, I get,
(y,x)=(y',x')
How can I show the function is injective?

Right so far. Can you conclude (x,y) = (x',y') from that?
 
  • #3
the ordered pairs are equal means that we can write y=y' and x=x' which in tern mean that
(x,y)=(x',y')

Is this fine.

Thank You.
 
  • #4
Yes, that's all there is to it.
 

Related to Injective Property of Rotation Function (x,y) to (y,x)

1. What is the Injective Property of Rotation Function?

The Injective Property of Rotation Function is a mathematical concept that describes the relationship between the input and output values of a rotation function. It states that for every unique input value, there is only one unique output value.

2. How is the Injective Property of Rotation Function different from other properties?

The Injective Property of Rotation Function is different from other properties, such as the Surjective and Bijective properties, because it focuses specifically on the uniqueness of the output values. In other words, the input values may be repeated, but the output values must always be distinct.

3. Can you give an example of a function that satisfies the Injective Property of Rotation Function?

Yes, the function f(x) = x^2 is an example of a function that satisfies the Injective Property of Rotation Function. For every unique input value, there is only one unique output value (i.e. f(2) = 4, f(3) = 9, etc.).

4. How is the Injective Property of Rotation Function useful in real-life applications?

The Injective Property of Rotation Function has many practical applications, such as in computer graphics and animation. It allows for the creation of smooth and seamless rotations of objects, as there are no duplicate output values that would cause visual glitches or errors.

5. Are there any limitations to the Injective Property of Rotation Function?

Yes, there are limitations to the Injective Property of Rotation Function. It only applies to functions that have a one-to-one mapping between input and output values. In other words, the function must be both injective and surjective (i.e. every output value must have a corresponding input value).

Similar threads

How should I use the Jacobi equation to show this?
    • Calculus and Beyond Homework Help
Replies
4
Views
766
Is It Possible to Invert a Homotopy?
    • Calculus and Beyond Homework Help
Replies
5
Views
403
How should I use the Jacobi equation to determine the nature of this?
    • Calculus and Beyond Homework Help
Replies
5
Views
694
Lagrange multipliers understanding
    • Calculus and Beyond Homework Help
Replies
8
Views
565
Prove ##a + b = b + a## using Peano postulates
    • Calculus and Beyond Homework Help
Replies
3
Views
859
Prove ##(a+b)\cdot c=a\cdot c+b\cdot c## using Peano postulates
    • Calculus and Beyond Homework Help
Replies
3
Views
580
##G## is Injective ##\iff ## ## f ## is onto ## Y ## (Lambda Notation)
    • Calculus and Beyond Homework Help
Replies
3
Views
783
How should I show that ## B ## is given by the solution of this?
    • Calculus and Beyond Homework Help
Replies
2
Views
511
Determine whether ## S[y] ## has a maximum or a minimum
    • Calculus and Beyond Homework Help
Replies
18
Views
1K
Find limit of multi variable function
    • Calculus and Beyond Homework Help
Replies
10
Views
870

Hot Threads

Recent Insights

Back
Top