How to say that functions generate set members?

Question

In a paper I am writing, I have seven abstract statements and, corresponding to the first one, I have a theorem statement that says I will prove that there is a Pythagorean triplet for every pair of natural numbers $n,k$ using functions: $$A(n,k)=(2n-1)^2+2(2n-1)k$$ $$B(n,k)=2(2n-1)k+2k^2$$ $$C(n,k)=(2n-1)^2+2(2n-1)k+2k^2$$ developed from $my$ observation that all $interesting$ triplets (especially primatives) are members of distinct sets as shown in the sample below (where $n$ is the set number and $k$ is the element number within the set). In each triplet in each set, the difference between B and C is always $(2n-1)^2$ and the increment between values of A is always $2(2n-1)$. The theorem statement so far is: $$\forall n,k\in\mathbb{N},∃ A,B,C\in\mathbb{N}|A^2+B^2=C^2\land \left\{A,B,C\right\}\in \left\{ ? \right\}$$ and here I want to correct the syntax for saying each triplet (A,B,C) is a unique element of a distinct set of triplets.

$$\begin{array}{c|c|c|c|c|} \text{$Set_n$}& \text{$Triplet_1$} & \text{$Triplet_2$} & \text{$Triplet_3$} & \text{$Triplet_4$}\\ \hline \text{$Set_1$} & 3,4,5 & 5,12,13& 7,24,25& 9,40,41\\ \hline \text{$Set_2$} & 15,8,17 & 21,20,29 &27,36,45 &33,56,65\\ \hline \text{$Set_3$} & 35,12,37 & 45,28,53 &55,48,73 &65,72,97 \\ \hline \text{$Set_4$} &63,16,65 &77,36,85 &91,60,109 &105,88,137\\ \hline \end{array}$$ I have written the proof(s) for the $existence$ and $set$-$membership$ but how do I describe the set membership in the theorem statement? In other words what goes inside the brackets with the question mark $\left\{ ? \right\}$ to show that every $set$ of triplets is an element of a greater set?

Answer 1

I am not sure that I correctly understood your question, but I guess that you are asking how to state that to distinct pairs $(n,k)$ of natural numbers correspond distinct triples $(A(n,k),B(n,k),C(n,k))$. That is, formally saying, that a map $\Pi:\Bbb N\times \Bbb N$, $(n,k)\mapsto (A(n,k),B(n,k),C(n,k))$ is injective. A different claim is that each Pythagorean triple $(A,B,C)$ equals $(A(n,k),B(n,k),C(n,k))$ for some natural $n$ and $k$. But if we are interested only in interesting triples then we don’t need this claim.

PS.

The theorem statement so far is: $$\forall n,k\in\mathbb{N},∃ A,B,C\in\mathbb{N}|A^2+B^2=C^2\land \left\{A,B,C\right\}\in \left\{ ? \right\}$$

When I was a schoolboy, I also tried to use such overformalized notation working with a close problems in order to be more cool. Now I know that it is better to formulate claims to be easy to read and understand, with more words, if needed. It is more professional, clear, and error-safe.

Answer 2

Give a definition:$$ \forall n\in \Bbb Z^+ \,\;[\, S_n=\{(A,B,C)\in (\Bbb Z^+)^3\,: A^2+B^2=C^2\, \land\, C-B=(2n-1)^2\}\,].$$

Then you have a lemma: $$\forall n\in \Bbb Z^+\, (S_n\ne \emptyset).$$ Proof: If $x\in \Bbb Z^+$ and $y=x+2n-1$ then $(y^2-x^2, 2xy,y^2+x^2)\in S_n.$

It is a very ancient result that every primitive Pyth. triplet $(A,B,C)$ satisfies $[\;\{A,B\}=\{y^2-x^2, 2xy\}\land z=y^2+x^2\; ]$ for a unique $(y,x)\in \Bbb Z^2....$ and that $x,y$ are co-prime and that $y-x$ is odd.