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afirican
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How do I prove that Q under addition is not isomorphic to R+ under multiplication?
afirican said:Isn't it f(x) = exp(x) a bijection between Q and R+?
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.
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.
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.
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.
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.