Solution
= Solution
Because $P(\mathbf x)\subseteq P(\mathbf x)$, the relation is <reflexive relation>[reflexive]. If $\mathbf x\preceq\mathbf y$ and $\mathbf y\preceq\mathbf z$, then
$$
P(\mathbf x)\subseteq P(\mathbf y)\subseteq P(\mathbf z),
$$
so it is <transitive relation>[transitive]. It is not <symmetric relation>[symmetric]: for
$$
\mathbf x=(1,0,\ldots,0),
\qquad
\mathbf y=(1,1,0,\ldots,0),
$$
one has $\mathbf x\preceq\mathbf y$ but not $\mathbf y\preceq\mathbf x$.