Recursive definition of the relation greater than on N X N

Question

Give a recursive definition of the relation greater than on N X N using the successor operators s?

I started this question throw this way:

basis: (1,0) ∈ N x N

could someone help me in recursive step?

thanks

Answer 1

If $(n,m) \in \ >$ then

$(S(n),m) \in \ >$ and

If $n \neq S(m)$

$(n,S(m)) \in \ >$

Answer 2

How about: if $(a,b) \in >$ then $(a+1,b)\in >$ and $(a+1,b+1)\in >$. (Here I am using $(x,y)\in >$ to signify $x>y$.)

Answer 3

My favorite recursive definition of $>$ for natural numbers is:

$(0,n) \notin \,>$ and

$(m+1,n)\in\,>\iff [(m,n)\in\,>\text{ or }m=n]$.