A function f: Z x Z is defined as f(m, n) = 3n - 4m. Verify whether this function is injective and whether it is surjective.
A function f: Z x Z is defined as f(m, n) = 3n - 4m. Verify whether this function is injective and whether it is surjective.
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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A function f: Z x Z is defined as f(m, n) = 3n - 4m. Verify whether this function is injective and whether it is surjective.
Expert Solution
Step 1
f: Z x Z is defined as f(m, n) = 3n - 4m
So, f(m, n) = f(a, b)
implies, 3n - 4m = 3b - 4a
i.e. 3(n - b) = 4(m - a), holds for all a, b, m, n in Z
But, gcd(3,4) = 1
So, the above equation holds for all a, b, m, n implies that, n - b = 0 = m - a
So, n = b &, m = a
So, (m, n) = (a, b)
So, f(m, n) = f(a, b) implies, (m, n) = (a, b)
So, f is injective.
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