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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?


asked in Set Theory & Algebra by Boss (24.9k points)
edited by | 94 views
a) one to one

b) onto and many to one

$(a)$ $f(m,n)=2m-n$ 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.

1st is not one to one check my comment
ok, yes thanks

that means, 2 numbers subtraction or addition or multiplication or division cannot be one to one

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