How to Prove a Function Is Onto in Mathematics

TL;DR

To prove that the function f(m, n) = m + n is onto, show that for every integer B, you can find integers M and N such that M + N equals B. Specifically, use M = 2B and N = -B, which satisfies the condition that f(M, N) = B, demonstrating that the function is surjective.

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

Key Insights

• 😫 The German word "zahl" meaning numbers is the reason why Z is used to denote the set of integers.
• 🖐️ Surjectivity plays a crucial role in proving the function's properties.
• 💦 The scratch work involves finding appropriate ordered pairs to satisfy the function's conditions.
• 🛀 The formal proof follows the structure of assuming B in Z, showing the existence of A, and verifying F of A equals B.
• ❓ Understanding the definition of surjectivity is fundamental to solving problems related to function properties in mathematics.
• ❓ The proof establishes the surjectivity or "on-2" property of the function.
• 😒 The use of scratch work is essential in developing a formal proof.

Questions & Answers

Q: What is the definition of a surjective function?

A function F from a domain A to a co-domain 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.