Proving Partial Order of R1 & R2 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.