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
Question

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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