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## Homework Statement

Help with either of these problems would be great.

1. Suppose R is a partial order on A and [tex]B\subseteq A[/tex]. Prove that [tex]R \cap \left(B\times B\right)[/tex] is a partial order on B.

2. Suppose R

_{1}is a partial order on A

_{1}, R

_{2}is a partial order on A

_{2}, and [tex]A_1 \cap A_2 = \emptyset[/tex]

Prove that [tex]R_1 \cup R_2[/tex] is a partial order on [tex]A_1 \cup A_2[/tex]

## The Attempt at a Solution

1. I'm confused what I am supposed to do with [tex]R \cap \left(B\times B\right)[/tex]...

2. I know that a partial order is a relation that is reflexive, antisymmetric, and transitive, so I would think that I would have to prove that [tex]R_1 \cup R_2[/tex] is reflexive, symmetric, and transitive on [tex]A_1 \cup A_2[/tex]. I'm able to prove that [tex]R_1 \cup R_2[/tex] is reflexive by supposing that x is an arbitrary element of [tex]A_1 \cup A_2[/tex] and then using the fact that R

_{1}and R

_{2}are reflexive. I can't figure out how to prove the antisymmetric and transitive parts though.

Thanks.