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Relation proof

  • Thread starter Testify
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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 R1 is a partial order on A1, R2 is a partial order on A2, 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 R1 and R2 are reflexive. I can't figure out how to prove the antisymmetric and transitive parts though.

Thanks.
 

Answers and Replies

  • #2
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(1) If [tex]B\subset A[/tex], then [tex]B\times B\subset A\times A[/tex]. 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 [tex]A_1,A_2[/tex] are disjoint to see that no element of [tex]A_1[/tex] is related to any element from [tex]A_2[/tex], and go from there.
 

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