- #1
afirican
- 5
- 0
How do I prove that Q under addition is not isomorphic to R+ under multiplication?
Prove that Q under addition is not isomorphic to R+
- Context: Graduate
- Thread starter afirican
- Start date
-
- Tags
- Addition
Click For Summary
Discussion Overview
The discussion centers on the question of whether the group of rational numbers under addition (Q) is isomorphic to the group of positive real numbers under multiplication (R+). Participants explore various aspects of group isomorphism, including bijective correspondence and the properties of specific functions proposed as potential isomorphisms.
Discussion Character
- Debate/contested
Main Points Raised
- One participant questions how to prove that Q under addition is not isomorphic to R+ under multiplication.
- Another participant asserts that the groups cannot be isomorphic due to a lack of bijective correspondence.
- There is a proposal that the function f(x) = exp(x) could serve as a bijection between Q and R+, which is later challenged.
- A participant argues that for f(x) = exp(x) to be an isomorphism, it must be onto, implying that it fails to meet this criterion.
- One participant seeks clarification on why f:Q -> R+, f(x) = exp(x) is not onto, suggesting that for every positive real number r, there exists a corresponding rational number r' such that r = exp(lnr).
- Another participant raises a question about the nature of irrational numbers and their relation to Q and R+.
- One participant suggests that demonstrating any map between Q and R+ is not onto may suffice to show they are not isomorphic.
- A later reply indicates that simply examining the two sets Q and R+ reveals that they cannot be isomorphic, referencing a prior comment by another participant.
Areas of Agreement / Disagreement
Participants express differing views on the potential for a bijection between Q and R+, with some arguing against the possibility of an isomorphism based on the properties of the groups involved. The discussion remains unresolved regarding the specifics of proving non-isomorphism.
Contextual Notes
Limitations include the need for clarity on the definitions of isomorphism and onto functions, as well as the implications of irrational numbers in the context of the proposed mappings.
Similar threads
- · Replies 4 ·
Undergrad
Do Metric Tensors Always Have Inverses?
- · Replies 8 ·
- · Replies 4 ·
- · Replies 5 ·
- · Replies 5 ·
- · Replies 31 ·
- · Replies 21 ·
- · Replies 8 ·
- · Replies 5 ·
- · Replies 19 ·