[injective and surjective fns] How do I test for injectivity and surjectivity on functions with multiple variables, like these?

4 viewed last edited 1 year ago
Peyton Tran
The book merely asked if these functions are surjective, not if they are injective. I want to test for both regardless (doing this for self-studies and have no way to check work). Could someone tell me how to test for injectivity & surjectivity on functions with multiple variables like these? Function maps: ZxZ->Z. (just picking one from random...) f(m,n)=|m|-|n| For injectivity I know f is injective iff f(x)=f(y)-> x=y How do I set this up for this problem to test for injectivity? |m| - |n| = |a| - |b|? What do I solve for, to show if it's injective here? Same for surjectivitity what do I need to solve for? I usually would find the inverse and inspect to see if any of the elements are outside the codomain. if I attempt that I get, |m| = |n| and then we have f^-1 (m,n)= 0 and sub'ing in I get |n| - |n| = 0. But that isn't making sense to me. I can do these sort of problems well whenever the function contains one variable or so. Confused on these sort of problems (the book doesn't really ask much of these sort off problems with multiple variables but want to know for my own understanding).