Proving R1 ∪ R2 is a partial order on A1 ∪ A2

[Testify]

Homework Statement

Help with either of these problems would be great.

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

2. Suppose R₁ is a partial order on A₁, R₂ is a partial order on A₂, and $$A_1 \cap A_2 = \emptyset$$

Prove that $$R_1 \cup R_2$$ is a partial order on $$A_1 \cup A_2$$

The Attempt at a Solution

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

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 $$R_1 \cup R_2$$ is reflexive, symmetric, and transitive on $$A_1 \cup A_2$$. I'm able to prove that $$R_1 \cup R_2$$ is reflexive by supposing that x is an arbitrary element of $$A_1 \cup A_2$$ and then using the fact that R₁ and R₂ are reflexive. I can't figure out how to prove the antisymmetric and transitive parts though.

Thanks.

 

[Tinyboss]

(1) If $$B\subset A$$, then $$B\times B\subset A\times A$$. Make sure you understand how a relation is defined in terms of the cartesian product of the base set with itself, and this should make sense immediately.

(2) Use the fact that $$A_1,A_2$$ are disjoint to see that no element of $$A_1$$ is related to any element from $$A_2$$, and go from there.