Trouble w/ injections, surjections, bijections

  • Thread starter semidevil
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In summary, the definition says that a function maps real numbers to natural numbers. The statement says that if there is no map from N to R, then there must be a bijection between the two. The problem is that it is not obvious how to prove that [0,1] is in bijection with R.
  • #1
semidevil
157
2
ok, from the definition, and drawing a picture, I can understand what all these mean. And when it comes to solving problems, I can solve them too. i.e 2x/1-x = 2y/y-1. easy, I know how to do it.

But when it comes ot applying it to finite/infinite sets, I don't know how to start. i.e, is there a bijection from the positive real number to the set of natural numbers.

ok, so positive real numbers imply 0, 1, 2,...n, and also everything in between.

natural number imply 1, 2, ...n.

so what does this mean? what does it mean if there is a bijection between these two? injection? surjection?

how do I tink this through?
 
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  • #2
Let P be the set of Positive real numbers and N is ofcourse the set of natural numbers

Is there a function f:P->N such that every element of P is mapped to exactly one element of N and every element of N is an image of an element of P??

Read through the definitions of function and injections , surjection and bijection and read through my statement ...

Then once u have understood what i have said, give us your initial thoughts that is what do u think should be the answer to the question and why?

-- AI
 
  • #3
To be honest I don't think that's going to help much for this particular question. You're going to have to use some ingenuity to show that no map from N to R is a bijection.

Any such map means you can list the real numbers x(1), x(2), x(3),... and get all of them in the list. Now you've got to show that there can be no such list by finding one not on the list. Hint, can you find a number different from x(n) in some way for all n?
 
  • #4
i wanted to skip the diagonal part atleast for the initial stages ... seeking an idea as to how much the person has thought upon the problem ...

-- AI
 
  • #5
1 isn't that obvious until you prove [0,1] is in bijection with R

and

3. is false.
 
  • #6
All right, message deleted.
 

1. What is the difference between injections, surjections, and bijections?

Injections, surjections, and bijections are all types of functions in mathematics. An injection is a function that maps each element in the domain to a unique element in the codomain. A surjection is a function that maps at least one element in the codomain to each element in the domain. A bijection is a function that is both an injection and a surjection, meaning it maps each element in the domain to a unique element in the codomain and also has a one-to-one correspondence between the domain and codomain.

2. How do you determine if a function is an injection, surjection, or bijection?

To determine if a function is an injection, you can use the horizontal line test. If a horizontal line intersects the graph of the function at more than one point, then the function is not an injection. To determine if a function is a surjection, you can check if every element in the codomain has at least one preimage in the domain. To determine if a function is a bijection, you can use both the horizontal line test and check for a one-to-one correspondence between the domain and codomain.

3. Can a function be both an injection and a surjection but not a bijection?

Yes, a function can be both an injection and a surjection but not a bijection. This means that the function maps each element in the domain to a unique element in the codomain and also has at least one element in the codomain that is not mapped to by any element in the domain.

4. Why are injections, surjections, and bijections important in mathematics?

Injections, surjections, and bijections are important in mathematics because they help us understand the relationship between two sets. Injections and surjections are used to prove the existence of a bijection, which is a special type of function that has important properties and is useful in many areas of mathematics, including set theory, abstract algebra, and topology.

5. How are injections, surjections, and bijections used in real life?

Injections, surjections, and bijections are used in many real-life applications, including computer science, economics, and biology. For example, in computer science, bijections are used in encryption algorithms to ensure that each input has a unique output. In economics, injections are used to model the relationship between supply and demand, and surjections are used to model the distribution of wealth. In biology, bijections are used to model the relationship between the genotype and phenotype of an organism.

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