Question

Determine whether the following functions $f$ are well defined: (a) $f: \mathbb{Q} \rightarrow \mathbb{Z}$ defined by $f(a / b)=a$. (b) $f: \mathbb{Q} \rightarrow \mathbb{Q}$ defined by $f(a / b)=a^{2} / b^{2}$.

Name: Problem 5, Chapter 0: Abstract Algebra
Uploaded: 2023-09-16T22:45:56-08:00
Duration: 2 min 47 s
Channel: Brandon Collins
Description: Determine whether the following functions f are well defined: (a) f: ℚ →ℤ defined by f(a / b)=a. (b) f: ℚ →ℚ defined by f(a / b)=a^2 / b^2.

   Determine whether the following functions $f$ are well defined:
(a) $f: \mathbb{Q} \rightarrow \mathbb{Z}$ defined by $f(a / b)=a$.
(b) $f: \mathbb{Q} \rightarrow \mathbb{Q}$ defined by $f(a / b)=a^{2} / b^{2}$.

Abstract Algebra

David S. Dummit,… 3rd Edition

Chapter 0, Problem 5

Instant Answer

Solved by Expert Brandon Collins

09/16/2023

Step 1

Consider the rational number $a/b$ where $a$ and $b$ are integers with no common factors other than 1 (i.e., $a/b$ is in lowest terms). Now, suppose there is another representation of the same rational number, say $c/d$, where $c$ and $d$ are integers with no Show more…

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Determine whether the following functions $f$ are well defined: (a) $f: \mathbb{Q} \rightarrow \mathbb{Z}$ defined by $f(a / b)=a$. (b) $f: \mathbb{Q} \rightarrow \mathbb{Q}$ defined by $f(a / b)=a^{2} / b^{2}$.

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Key Concepts

Well-Defined Functions

A function is well defined if the output is uniquely determined for each input, regardless of how that input is represented. When dealing with sets like the rationals, which can have multiple representations for the same element, it is imperative that applying the function to any equivalent representation results in the same output.

Equivalence of Representations

Rational numbers are equivalence classes of pairs of integers with nonzero denominators, meaning that different pairs (a, b) can represent the same rational number. A function that takes a representation as its input must ensure that its value remains consistent across all equivalent representations, or else it is not well defined.

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