$\lbrace 2n+3m+1:n,m\in N\rbrace$ is the set of all positive integers except for $0$ and $2$. I need to know how to write its inductive definition.
This is part of the introduction on learning how to develop recursive functions using lambda calculus. I can do some of them but on others, such as this one, I get lost. How do you handle the multiple variables. Please explain how you got your answer as well since an answer doesn't do me much good if I don't know how to get it.
Here are two of the ones I know how to do.
$\lbrace 3n+2: n\in \mathbb N\rbrace$
Top Down: $n = 2; n - 3 \in S$
Bottom up: $2 \in S$; if $n \in S$, then $(n + 3) \in S$
Rule of Inference: $2 \in S$; if $n \in S$, then $(n+3) \in S$
$\lbrace(n,2n+1): n\in \mathbb N\rbrace$
Top Down: $(n,m)=(0,1);(n-1,m-2) \in S$
Bottom up: $(0,1) \in S$; if $(n,m) \in S$, then $(n+1,m+2) \in S$