Please, give me an example f:NNxxNN\toNN f is a bijection ?
2 Answers
Please see below
Explanation:
Here ,
We take ,
f(x,x)=(x+x)/2=x ,x inNN
f(1,1)=(1+1)/2=1
f(2,2)=(2+2)/2=2
f(3,3)=(3+3)/2=3
f(4,4)=(4+4)/2=4
color(white)(..)vdots vdotscolor(white)(...)vdotscolor(white)(....)vdotscolor(white)(.)vdots
f(n,n)=(n+n)/2=n , where ,n in NN
color(white)(..)vdots vdotscolor(white)(...)vdotscolor(white)(....)vdotscolor(white)(.)vdots
Co-domain of
. Range ofcolor(blue)(f=R_f=NN
Explanation:
I will assume that
Consider the triangular numbers:
T_0 = 0
T_1 = 1
T_2 = 1+2 = 3
T_3 = 1+2+3 = 6
...
T_n = sum_(k=1)^n k = 1/2n(n+1)
We can define:
f(m, n) = T_(m+n)+n = 1/2(m+n)(m+n+1)+n
color(white)(f(m, n)) = 1/2(m^2+n^2+2mn+m+3n)
This tells you the
(0, 0), (1, 0), (0, 1), (2, 0), (1, 1), (0, 2), (3, 0), (2, 1), (1, 2), (0, 3),...
which enumerates the points of
Footnote
If you want a bijection from
g(m, n) = f(m-1, n-1)+1
and simplify.