Let $\mathrel{R_1}$ and $\mathrel{R_2}$ be relations on $\mathbb N$ defined by
$x\mathrel{R_1}y$ if and only if $y = a + x$ for some $a \in\mathbb N_0$.
$x\mathrel{R_2}y$ if and only if $y = xa$ for some $a \in\mathbb N$
For all $x; y\in\mathbb N.$
Also $\mathbb N_0$ denotes all integers $x\ge 0$, while $\mathbb N$ denotes all integers $x\ge 1$.
There are four different things I want to prove with this.
I want to show that $\mathrel{R_1}$ is a partial order on $\mathbb N$.
I want to show that $\mathrel{R_2}$ is a partial order on $\mathbb N$.
I want to show that $\mathrel{R_2}\subseteq\mathrel{R_1}$.
Finally I want to show that $\mathrel{R_1}$ is a total order.
For the first proof that $\mathrel{R_1}$ is a partial order I need to show that $\mathrel{R}$ is Reflexive, Anti-Symmetric, and Transitive, and the same goes for R2.
$\mathrel R$ is reflexive: for all $a\in A, a\mathrel Ra$.
$\mathrel R$ is antisymmetric: for all $a,b\in A$, if $a\mathrel Rb$ and $b\mathrel Ra$, then it must be the case that $a=b$.
$\mathrel R$ is transitive: for all $a,b,c\in A$, if $a\mathrel Rb$ and $b\mathrel Rc$, then $a\mathrel Rc$.
Unfortunately my problem is every time I write a proof I fail miserably because I don't have anything to base off of. I understand how it should work I just don't understand how to correctly write it.
In order to show that $\mathrel{R_2}\subseteq\mathrel{R_1}$ this means that everything that is in $\mathrel{R_2}$ must be in $\mathrel{R_1}$?
The last part has me very confused because I have to show that $\mathrel{R_1}$ is a total order. I think this means that I have to show that it is Anti-Symmetric, Transitive, and a totality. I wouldn't know where to start for this.
What I need overall is help with forming with these proof's and to see if someone could give me a guide line sort of basis to perform my work off of.
Ḧints: