## Injective mapping

Show that the mapping $$f: \mathbb N\times \mathbb N\to \mathbb N$$ defined by the equation $$\displaystyle f(x,y)=\frac{(x+y)(x+y+1)}{2}+y$$ is injective.

• #### Hint

First try to arrange the pairs $$(x,y)$$ such that the value of the function $$f$$ increases.

• #### Resolution

Without loss of generality we first consider the case $$(x+y)<(x'+y')$$. Then

$f(x,y)=\frac{(x+y)(x+y+1)}{2}+y=\frac12(x^2+2xy+y^2+x+3y) <\frac12(x^2+2xy+y^2+3x+3y)=$ $=\frac12(x+y+1)(x+y+2)\le\frac12(x'+y')(x'+y'+1) <\frac{(x'+y')(x'+y'+1)}{2}+y=f(x',y')$ .

If $$(x+y)=(x'+y')$$ and $$y<y'$$ we have $$f(x,y)<f(x',y')$$ immediately.

In the case that $$(x+y)=(x'+y')$$ and $$y=y'$$ then also $$x=x'$$. So the function $$f$$ is injective.