● Covered by 1 source · 1 reportLow impact1 neutral
Mathematician identifies error in Dummit and Foote's Abstract Algebra textbook
Key points
- A proof exercise in "Abstract Algebra" was found to be incorrect.
- The statement claims injectivity is equivalent to having a left inverse.
- The counterexample uses an empty domain and a non-empty codomain.
- The error was identified using the formal verification tool Rocq.
Discovery of a Mathematical Error
During a formalization effort using Rocq, a mathematician identified an error in the first proof exercise of Dummit and Foote's textbook, "Abstract Algebra." The exercise posits that a function is injective if and only if it possesses a left inverse. This statement was found to be false.
The Counterexample
The counterexample involves a function f from an empty set A to a non-empty set B. Such a function is vacuously injective because there are no distinct inputs to map to the same output. However, this function does not have a left inverse, as no function can map from the non-empty set B to the empty set A, thus disproving the exercise's claim.
Role of Formal Verification
The error was discovered due to the rigorous nature of formal verification with Rocq. The tool's insistence on provable statements prevented the mathematician from completing the proof as stated, leading to the investigation of the proposition's truthfulness. This demonstrates how formal methods can uncover subtle logical flaws that might be overlooked in traditional paper-based proofs.
Prior Acknowledgment
The identified error is already listed in the book's errata, indicating that the inaccuracy had been previously recognized by the authors or other readers. This confirms the validity of the mathematician's finding.
✨ This summary was generated by AI from the outlets' reporting listed below. It is not independently verified and may contain errors — check the original sources. How BrevFeed works →
The daily brief
One email each morning: the day's tech stories, clustered across outlets and summarized. No account needed.
One email a day. Unsubscribe in one click, any time.
Today's brief
Spend a few minutes, get the whole day. Every topic's top stories in one hands-free rundown — listen, watch, or read the transcript.
▶ Play today's brief
New every morning, and the back catalogue is archived by date.
Reporting from
A mathematician discovered an error in the first proof exercise of Dummit and Foote's "Abstract Algebra" textbook while formalizing it in Rocq. The exercise incorrectly states that a function is injective if and only if it has a left inverse, which is disproven by a corner case involving empty sets. This finding highlights the utility of formal verification tools in identifying subtle mathematical inaccuracies.