# Create an injection from {a^2+b^2

• aaaa202
In summary: Problems like this one are caused by not thinking about these things properly. In summary, the problem is that there are many combinations of a,b that give the same number, so it is not possible to create an injection from {a^2+b^2 l a,b∈Q} to {{a+b l a,b∈Q}}.
aaaa202

## Homework Statement

Is it possible to create an injection from {a^2+b^2 l a,b$\in$Q} to {{a+b l a,b$\in$Q}

## Homework Equations

I am not sure about this. The problem is that many combinations of a,b can yield the same number, so how do I tackle this obstacle?

## The Attempt at a Solution

Consider the definition of injection.

the definition says:

x1≠x2 => f(x1)≠f(x2)

But the problem is as said that many combinations give rise to the same numbers in the set.

Then perhaps it is not possible!

it certainly should be, since the two sets have same cardinality. I just need help with coming up with a well defined function that does it. Problem is you can't really say something like f(a^2+b^2)=lal+lbl because of the earlier stated.

aaaa202 said:
it certainly should be, since the two sets have same cardinality.
Is this necessarily true?

aaaa202 said:
I just need help with coming up with a well defined function that does it.

It might simplify the problem somewhat to observe that ##\{a+b \, | \, a,b\in \mathbb{Q}\}## is simply ##\mathbb{Q}##.

I've just looked at a couple of your other posts like this one where again you seem to be confused about what you mean by {a^2+b^2 l a,b∈Q}. Can you see that all elements of this set are in ## \mathbb{Q} ##, and that a and b are only relevant in selecting which elements of ## \mathbb{Q} ## are elements of this set? You can't then define a function f(a,b) and expect it to have any meaning in relation to this set without defining how you get the unique pair (a,b) from ## x \in \mathbb{Q} ##.

Before you try to think about jections (I hate that word) you need to think more clearly about the membership of sets and the domain, co-domain and range of relations between them.

Last edited:

## 1. How do you create an injection from a^2+b^2?

To create an injection from a^2+b^2, you would need to follow a specific set of steps. First, you would need to define the function f(x) = a^2+b^2. Then, you would need to show that the function is one-to-one by proving that f(x1) = f(x2) implies x1 = x2. Finally, you would need to show that the function is onto by proving that for every y in the range of f(x), there exists an x in the domain of f(x) such that f(x) = y. This will demonstrate that f(x) is a bijective function, and therefore an injection.

## 2. What is the purpose of creating an injection from a^2+b^2?

The purpose of creating an injection from a^2+b^2 is to demonstrate that the function is a one-to-one mapping, which means that no two distinct elements in the domain will map to the same element in the range. This is important in many areas of mathematics and science, as it allows us to prove the existence of a unique solution to a problem.

## 3. Can you provide an example of an injection from a^2+b^2?

One example of an injection from a^2+b^2 is the function f(x) = x^2. This function maps every real number in the domain to a unique real number in the range. For example, f(2) = 4 and f(3) = 9, showing that no two distinct numbers in the domain map to the same number in the range.

## 4. How does an injection from a^2+b^2 differ from other types of functions?

An injection from a^2+b^2 differs from other types of functions in that it is both one-to-one and onto, meaning that every element in the range has a unique pre-image in the domain. This is not the case for all functions, as some may not be one-to-one, meaning that two distinct elements in the domain can map to the same element in the range.

## 5. What are some real-world applications of creating an injection from a^2+b^2?

Creating an injection from a^2+b^2 has many real-world applications, particularly in fields such as computer science, engineering, and physics. For example, in computer science, creating an injection allows us to prove the efficiency and correctness of algorithms, while in physics, it can help us solve problems involving motion and energy. In engineering, injections can be used to optimize designs and improve the reliability of systems.

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