Given the function: $f:(a,b)\in\mathbb{Z}\times\mathbb{Z}\longrightarrow ab^2\in\mathbb{Z}$ What can you say about injectivity and surjectivity?
This is not injective. I have easily find two elements that broke injectivity definition. For instance $(1,2)$ is different from $(1,-2)$ but their images is 4 in both case. This is surjective cause, every couple of number $(a,b)\in\mathbb{Z}$ have a correspective inside the codomain. But, what if I can't easily find the couple of element that broke the rules? What mathematic process can I use to confirm my results?
Considering in $\mathbb{Z}\times\mathbb{Z}$ the following relation: $(a,b)\Sigma(c,d)\Leftrightarrow a\leq c\quad\text{AND}\quad b\leq d$ Prove that $\Sigma$ is not a total order relation and determine min, Max, minimal and maximal element. I'm able to responde this question when I can sketch Hasse digram, so when I have finite set of element, but, what in this case?
Help me, best regards MC