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14. Determine whether f: ZxZ → Z is onto if a) f(m, n) = 2m - n. b) f(m, n) = m² = n². c) f(m, n) = m +n + 1. d) f(m, n) = \m| – |n|. e) f(m, n) = m² - 4. —

14. Determine whether f: ZxZ → Z is onto if a) f(m, n) = 2m - n. b) f(m, n) = m² = n². c) f(m, n) = m +n + 1. d) f(m, n) = \m| – |n|. e) f(m, n) = m² - 4. —

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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14a

**Problem 14: Determine Whether the Function is Onto** Let \( f: \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z} \). You need to determine whether the function \( f \) is onto for the following cases: a) \( f(m, n) = 2m - n \). b) \( f(m, n) = m^2 - n^2 \). c) \( f(m, n) = m + n + 1 \). d) \( f(m, n) = |m| - |n| \). e) \( f(m, n) = m^2 - 4 \). --- Explanation for educational purposes: An "onto" function, or surjective function, means that for every element \( z \) in the codomain \( \mathbb{Z} \), there is at least one element \((m, n)\) in the domain \(\mathbb{Z} \times \mathbb{Z}\) such that \( f(m, n) = z \). You will need to evaluate each given function to see if this property holds.
Transcribed Image Text:**Problem 14: Determine Whether the Function is Onto** Let \( f: \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z} \). You need to determine whether the function \( f \) is onto for the following cases: a) \( f(m, n) = 2m - n \). b) \( f(m, n) = m^2 - n^2 \). c) \( f(m, n) = m + n + 1 \). d) \( f(m, n) = |m| - |n| \). e) \( f(m, n) = m^2 - 4 \). --- Explanation for educational purposes: An "onto" function, or surjective function, means that for every element \( z \) in the codomain \( \mathbb{Z} \), there is at least one element \((m, n)\) in the domain \(\mathbb{Z} \times \mathbb{Z}\) such that \( f(m, n) = z \). You will need to evaluate each given function to see if this property holds.
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