I know an injective map from $\mathbb{N}\times \mathbb{N} \to \mathbb{N}$ given by the following explicit map:
$$(m,n) \to \frac{(m+n)(m+n-1)}{2}+m$$
Now if I take set $A=\{1,2^2,3^2,\cdot \cdot\cdot\} $ collection of all perfect squares. And define a map $f:A\times A \to \mathbb{N} $ with the help of earlier map from $\mathbb{N}\times \mathbb{N} \to \mathbb{N}$ as follows:
$$f(m^2,n^2)= \frac{(m+n)(m+n-1)}{2}+m$$
I am curious to know is the map $f$ injective. I have a feeling that this map $f$ is injective. But I am struggling to prove it or to find a counterexample.
Let $a,b,c,d \in A$, if $f(a,b)=f(c,d)$ then,
$$g(\sqrt{a},\sqrt{b})=g(\sqrt{c},\sqrt{d})$$
Where $g : \mathbb{N}\times \mathbb{N} \rightarrow \mathbb{N}$ denotes the injective map you defined at first.
Hence, as $g$ is injective we have $\sqrt{a} = \sqrt{c}$ and $\sqrt{b} = \sqrt{d}$, and $\sqrt{\cdot}$ is injective therefore, $a = c$ and $b = d$, hence $(a,b) = (c,d)$.
Therefore $f$ is injective.