Trouble w/ injections, surjections, bijections

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Discussion Overview

The discussion revolves around the concepts of injections, surjections, and bijections, particularly in the context of mapping positive real numbers to natural numbers. Participants explore the implications of such mappings and the challenges associated with establishing bijections between finite and infinite sets.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant expresses confidence in solving problems related to injections and surjections but struggles with applying these concepts to finite and infinite sets.
  • Another participant questions whether a function exists that maps every positive real number to exactly one natural number and vice versa, prompting a deeper examination of the definitions involved.
  • A suggestion is made that ingenuity is required to demonstrate that no bijection exists from natural numbers to real numbers, hinting at the diagonal argument as a potential method.
  • One participant indicates a desire to avoid complex arguments initially and seeks to gauge the understanding of the problem by others.
  • Another participant asserts that proving a bijection between the interval [0,1] and the real numbers is not immediately obvious, suggesting that some foundational proofs may be necessary.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the existence of a bijection between the positive real numbers and natural numbers, with some arguing against it and others seeking clarification on the definitions and implications.

Contextual Notes

The discussion includes unresolved assumptions about the nature of mappings between infinite sets and the definitions of injections, surjections, and bijections. The participants' varying levels of familiarity with these concepts may influence their contributions.

Who May Find This Useful

This discussion may be useful for individuals studying set theory, particularly those interested in the properties of infinite sets and the relationships between different types of functions.

semidevil
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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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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
 
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?
 
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
 
1 isn't that obvious until you prove [0,1] is in bijection with R

and

3. is false.
 
All right, message deleted.
 

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