Let $S$ be a non-empty set. Let $F \subset 2^{Sx\mathbb{Z}}$ be the set of all relations $G$ between S and $\mathbb{Z}$, where G is a graf.
1) Define a relation on $F$ which satisfy that a graf $G_1$ is in relation to another graf $G_2$, if and only if $g_1(s)≤g_2(s)$ for all $s \in S$. Let $G_i$ be the function belonging to the graf $G_i$ for $i=1,2,...$
2) Let $S=\mathbb{Z}$. Let $p_1=k^2-2k$ and let $p_2=2k^2-k+1$ Using the relation made in (1), evaluate if $p_1$ is in relation to $p_2$, and if $p_2$ is in relation to $p_1$ or if neither is true.
Here is what I've tried:
From what I've gathered I simple need to write down the relation.
Let $R$ be the relation on $F$. In the text we are told that
$$R=\{(g_1,g_2) \in F|g_1(s)≤g_2(s) \forall s\in S\}$$
Can that really be the answer?
I am uncertain how to do (2). I assume I somehow can check with the definition I have of a relation. The definition I have it as follows:
Let $S, T$ be sets. A relation $R$ between S and T is a subset of the cartesian product of S and T. If $(s,t) \in R$, then we say that $s \in S$ is the relation with $t \in T$.