Prove the Function is Onto: f(m, n) = m + n  Summary and Q&A
TL;DR
This video provides a proof that the function F of M n equals M plus n is surjective.
Key Insights
 š« The German word "zahl" meaning numbers is the reason why Z is used to denote the set of integers.
 šļø Surjectivity plays a crucial role in proving the function's properties.
 š¦ The scratch work involves finding appropriate ordered pairs to satisfy the function's conditions.
 š The formal proof follows the structure of assuming B in Z, showing the existence of A, and verifying F of A equals B.
 ā Understanding the definition of surjectivity is fundamental to solving problems related to function properties in mathematics.
 ā The proof establishes the surjectivity or "on2" property of the function.
 š The use of scratch work is essential in developing a formal proof.
Transcript
and this problem we're going to prove that this function from Z cross Z into Z given by F of M n equals M plus n is surjective so Z here is the set of integers so I believe the letter Z is used because of the German word sahlan and I think that means numbers in German so it's this set here negative 2 negative 1 0 1 2 and so on so this is the set of... Read More
Questions & Answers
Q: What is the definition of a surjective function?
A function F from a domain A to a codomain B is surjective if, for every element B in B, there exists an element A in A such that F of A equals B.
Q: Why is understanding the definition of surjectivity important for this proof?
Understanding surjectivity is crucial because the proof aims to show that for any element B in Z, there exists an ordered pair A in Z x Z such that F of A equals B.
Q: How is the scratch work done to determine the existence of A such that F of A equals B?
The scratch work involves taking a B in Z and finding appropriate integers M and N such that M plus N equals B. By assuming M equals 2B and N equals B, it is shown that F of 2B, B equals B.
Q: How is the formal proof structured to prove the surjectivity of the function?
The proof starts by assuming a B in Z and then shows the existence of an ordered pair A in Z x Z, satisfying the condition F of A equals B. By demonstrating that F of 2B, B equals B, it is concluded that the function is surjective.
Summary & Key Takeaways

The video explains the definition of surjective functions and the importance of understanding it for the proof.

The scratch work involves taking a B in Z and finding an ordered pair A in Z x Z such that F of A equals B.

The formal proof begins by assuming a B in Z, showing the existence of A, and verifying that F of A equals B.