Prove that Q under addition is not isomorphic to R+

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In summary, the conversation discusses the concept of isomorphism between Q and R+ as groups under addition and multiplication respectively. It is concluded that they cannot be isomorphic because they are not in bijective correspondence. The suggestion is made that any map between the two sets is not onto, indicating that Q and R+ are not isomorphic.
  • #1
afirican
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How do I prove that Q under addition is not isomorphic to R+ under multiplication?
 
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  • #2
They cannot be isomorphic as groups because they are not even in bijective correspondence as <insert one word to get the answer>
 
  • #3
Isn't it f(x) = exp(x) a bijection between Q and R+?
 
  • #4
afirican said:
Isn't it f(x) = exp(x) a bijection between Q and R+?

No an isomorphism must be onto.
 
  • #5
Why f:Q -> R+, f(x) = exp(x) is not onto?
For all r of R+, there exists r' = lnr in Q such that r = exp(lnr) = exp(r') = f(r'). Where do I go wrong?
 
  • #6
If r is irrational is r in Q? is er in R+?
 
  • #7
You're totally right. Then is there any way to show that Q and R+ are not isomorphic?
 
  • #8
I think I know the answer. If I say that any map between Q and R+ is not onto, is that enough?
 
  • #9
Yes, you just need to look at the two sets Q and R+ to see that the two groups cannot be isomorphic (as Matt grime indicated).
 

What is the definition of isomorphism?

Isomorphism is a mathematical concept that describes a one-to-one correspondence between two structures or systems. In the context of abstract algebra, isomorphism means that two algebraic structures have the same structure and can be mapped onto each other in a way that preserves their operations and properties.

What is Q under addition?

Q under addition refers to the set of rational numbers (fractions) with the operation of addition. This set includes all numbers that can be expressed as a ratio of two integers, such as 1/2, 3/4, and -5/9.

What is R+?

R+ refers to the set of positive real numbers, which includes all numbers that are greater than zero. This set does not include zero or any negative numbers.

Why is Q under addition not isomorphic to R+?

Q under addition and R+ are not isomorphic because they have different structures and properties. The set of rational numbers has a countably infinite number of elements, while the set of positive real numbers is uncountably infinite. Additionally, the operations of addition in these sets behave differently, as rational numbers have a unique additive inverse while positive real numbers do not.

Can you provide an example to prove that Q under addition is not isomorphic to R+?

Yes, consider the rational numbers 1/2 and 1/3. These numbers have a unique sum of 5/6 in Q, but there is no positive real number that can be added to itself to equal 5/6 in R+. This means that the operation of addition in Q under addition is not preserved in R+, indicating that these two structures are not isomorphic.

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