Given a partial ordering $R$ over a set $S$ is it true that for every $A\subseteq S$ that $R$ is also a partial ordering over $A$? I think so but I'm not sure.
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Yes, it is. The three defining properties of a partial order, reflexivity, antisymmetry and transitivity, contain only universal quantifiers and no existential quantifiers, and therefore can't be broken my removing elements from the set.