How to prove a function is onto?
Understand the Problem
The question is asking for the method or process to prove that a given function is onto (surjective). An onto function is one where every element in the codomain is mapped to by at least one element from the domain. To prove that a function is onto, we typically demonstrate that for every element in the codomain, there exists a corresponding element in the domain that maps to it.
Answer
Show for every y in B, there is an x in A such that f(x) = y.
To prove a function is onto, show that for every element y in the codomain B, there exists an element x in the domain A such that f(x) = y.
Answer for screen readers
To prove a function is onto, show that for every element y in the codomain B, there exists an element x in the domain A such that f(x) = y.
More Information
An onto function, also known as a surjective function, ensures that every element of the codomain is the image of at least one element from the domain.
Tips
Common mistakes involve not checking the condition for every element in the codomain, or assuming the function is onto without a rigorous proof.
Sources
- 6.4: Onto Functions - Mathematics LibreTexts - math.libretexts.org
- SplashLearn - Onto Function - splashlearn.com
- Proving a function is onto - StackExchange - math.stackexchange.com
AI-generated content may contain errors. Please verify critical information
