Partial ordering relation

Question

Consider the following system of sets and operations an them:

$(1)(Z,\geq ),$ where $'Z'$ is the set of an integer

$(2) (Z^{+},1),$where $'1'$ is the divisibility relation

$(3) (P(S),\subseteq ),$where $'P(S)'$ is the power set of $'S'$

Which of the above is/are partial ordering?

$A) (1)$ only

$B) (2)$ and $(3)$ only

$C) (1)$ and $(3)$ only

$D)$ All are partial ordering

Comments

i don't know how to prove this for all pairs of elements

let a and b are the elements, if (a,b) present in the relation means, a can divide b

it means b should be multiple of a ===> b = k.a ===>$\frac{b}{a}=k$

then (b,a) could be in the relation? (forget k=1, due to it leads reflexivity)

if (b,a) in the relation, then b should divide a ==> $\frac{a}{b}=p$

$\frac{a}{b}=p$

$\frac{a}{k.a}=p$

$\frac{1}{k}=p$ ===> $\frac{1}{k}$ can't be a integer except k=1, but note that our domain is integers only.

∴ b can't divides a ===> (b,a) can't be in the relation ===> given is anti-symmetric

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where ′P(S)′ is the power set of ′S′

here P(S) is only one set so how we can apply the subset relation on this

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@Shaik Masthan

In the question of (2)i.e. with divisibility operation ,Is Z-, i.e. set of -ve integers, is partial ordered?