Tagged Questions

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Ingredients for Proving that a set is bounded below

I am having a hard time to really understand the definition of bounded below. The definition states : Let S be a nonempty subset of real numbers, we say S is bounded below if there is a c ∈ ℝ ...
91 views

Defining a partial order on $A\times B$, given partial orders on $A$ and on $B$

Let $(A,\preceq_A)$ and $(B,\preceq_B)$ be partially ordered sets. Define $C = A \times B$ and define the relation $\preccurlyeq$ on $C$ to be $(a,b) \preccurlyeq$ $(a',b')$ if and only if ...
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Partitioning a totally ordered set into three subsets according to the order

Consider a set $S$, and a total order $R$ over that set. Part (a) Given some element $e \in S$, explain why it is possible to partition $S$ into the following three sets: S_1 = \{ x \in S ...
18 views

Diagonal Relation as a poset I - establishing the result by vacous truth

I have a problem with the logic behind the fact that, given a nonempty set X, the diagonal relation $D_X := \{ (x,x) : x \in X \}$ is a partial order on $X$. More specifically, my problem is with ...
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Problems with a proof that -in a linear order- a minimal element is the smallest element

I have a problem with a proof I found in Velleman's "How to prove it". This is sort of interesting, because it is the very first time I cannot see the structure of a proof presented in the book. The ...