Relations with partial order on a subset.

Question

Let R be a relation on the set X. For some $Y \subset X $ (a proper subset of $X$),

Let $R_Y $ be the relation on Y given by $(a,b) \in R_Y $ iff $ a,b \in Y $ and $ (a,b) \in R $

Prove that if R is a partial order on X then $R_Y $ is a partial order on Y

Answer 1

Hints:

This is mostly an exercise in satisfying the properties of a partial order:

A relation R is a partial order on a set $S$ if and only if the relation has the following properties:

It is reflexive: For every $a \in S$, $(a, a)\in R$.

It is antisymmetric: For every $a, b \in S,$ if $(a, b) \in R$ and $(b, a) \in R,$ then it must be the case that $a = b$.

It is transitive: For every $a, b, c \in S,\;$ if $(a, b) \in R$ and $(b, c)\in R$, then $(a, c) \in R$.

You are asked to prove that, if we are given that $R$ is a partially ordered relation on $X$, (meaning all the above properties hold for $R$ on $X$), then $R_Y$ is a partial order on $Y$ (i.e., all the properties of a partial order hold for $R_Y$ on Y, as well).

So assume the hypothesis, $R$ is a partial order on $X$, and $Y\subset X$, and use the fact that $$(a, b) \in R_Y \iff \big((a, b \in Y) \land ((a, b) \in R)\big).$$