proof about partial order

Question

Let R and S be two partial orders on a set X, and T is a relation on X such that aTb(i.e. a,b ∊ X) if and only if both aRb and aSb hold. Is T also a partial order on X? how to prove it?

Answer 1

Check each of the required properties for a relation to be a partial order: reflexivity, antisymmetry, and transitivity.

Here, I go through the proof that $T$ is transitive.

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Suppose $aTb$ and $bTc$. We ask whether or not this implies that $aTc$ is also true.

Well... by $aTb$ we know in particular that $aRb$. Further by $bTc$ we know in particular that $bRc$. Since $R$ is a partial order, we know that from $aRb$ and $bRc$ and that $R$ is transitive that $aRc$ is also true.

Similarly, we can show that $aTb$ and $bTc$ imply that $aSb$ and $bSc$ from which it follows that $aSc$.

Since both $aRc$ and $aSc$ are true, it follows that $aTc$ is also true.

Thus, the relation $T$ is indeed transitive.

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The remaining parts of the problem are just checking the other properties and they are all performed similarly. Use what you know about how $T$ is defined in terms of $R$ and $S$ and use the fact that $R$ and $S$ are both partial orders themselves and have the desired properties.