4 votes 4 votes How to test whether function is onto and one-to-one when function is in two variables? Determine whether below function $f:Z\,X\,Z\rightarrow\,Z$ is one-to-one, or onto or none? (a)$f(m,n)=2m-n$ (b)$f(m,n)=m^2-n^2$ Set Theory & Algebra kenneth-rosen discrete-mathematics set-theory&algebra functions + – Ayush Upadhyaya asked Sep 25, 2018 edited Mar 4, 2019 by Pooja Khatri Ayush Upadhyaya 562 views answer comment Share Follow See all 5 Comments See all 5 5 Comments reply srestha commented Sep 25, 2018 i edited by srestha Sep 25, 2018 reply Follow Share a) one to one b) onto and many to one 0 votes 0 votes Mk Utkarsh commented Sep 25, 2018 reply Follow Share $(a)$ $f(m,n)=2m-n$ is not one-to-one $f(2,2) = 2 = f(3,4)$ also for every $x \in Z$ there exist some m,n which generate x from $f(m,n)$, let x = 3, $2m - n = 3$ $2m = 3 + n$ now substitute any value of m and n that satisfies the equation so it is onto $(b)$ $f(m,n)=m^2 - n^2$ if x = 2 then there exist no m and n to satisfy this equation, $1^2 - 0 = 1$ and $2^2 - 1^2 = 3$ clearly not onto now $f(1,0)$ = 1 = $f(-1,0)$ so it is also not one to one. 3 votes 3 votes srestha commented Sep 25, 2018 reply Follow Share yes 1st one is one to one and onto 2nd one is not onto and many to one https://math.stackexchange.com/questions/4467/how-to-prove-if-a-b-in-mathbb-n-then-a1-b-is-an-integer-or-an-irratio 0 votes 0 votes Mk Utkarsh commented Sep 25, 2018 reply Follow Share 1st is not one to one check my comment 1 votes 1 votes srestha commented Sep 26, 2018 reply Follow Share ok, yes thanks that means, 2 numbers subtraction or addition or multiplication or division cannot be one to one 0 votes 0 votes Please log in or register to add a comment.