Question

12. From the given functions from Z × Z to Z, identify the onto functions. (Check all that apply.)

From the given functions from Z x Z to Z, identify the onto functions. (Check all that apply) Check All That Apply Am, n) 2m-n Am, n)= m2 - n2 Am, n) m+n+1 Am, n)= m - |n! Am, n) m2-4

Answer 1

(i)
F(m, n) 2m n
, the function is onto. For any $a\in \mathbb Z$, consider the point
[0,-a) EZx Z
, and
F(0,-a)a
, hence onto.

(ii)
F(m,n)= m2- n'
. Note that f is not onto. As there will not exists any
|(m, n) E Zx Z
, be such that
f(m, n) 2
. As if there exists then we have
n2-n22=> (m n)(m n) = 2
, since 2 is a prime and
(т - п) 2 —> (т + п)|2 оr (т - п)2 (т + п)(п
. Now note that either or . If m+n=2 then m-n=1 then we have m=1.5 a contradiction, similarly if m+n=-2 implies m-n=-1 and then m=-1.5 again contradiction. Similar checking can show m-n can not be 2 or -2.

(iii) $f(m,n)=m+n+1$ , f is onto as for any $a\in \mathbb Z$, $f(0,a-1)=a$.

(iv)
f(m, n)m
, is onto as for any $a\in \mathbb Z$, we have either
Πα.Οα
if $a\ge 0$ or $f(0,a)=a,\, a\ge 0.$

(v) $f(m,n)=m^2-4$. f is not onto as there will not exists
|(m, n) E Zx Z
, such that .

Feel free to comment if you have any doubts. Cheers!