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# Alex Kreitzberg

• Chris Nolan #29: The answer to all your questions is yes. For your example: Let poset A = ( {a, b}, { (b, a), (a, a), (b, b) } ) and B = ( {0, 1}, { (0, 1), (0, 0), (1, 1) } ) In abuse of notation, I'll also write A = {a, b}, B = {0, 1}, and $$\le… • Chris Nolan#19 If all the comparable elements of two sets are equal, then the sets are equal. But posets also include an order, and the order of two sets could be different, even though the elements are the same. Now, if by poset, you mean posets … • Puzzle 14 Checking some concrete values, \(2(1) \leq 3, 2(2) \not \leq 3, 2(2) \leq 5, 2(3) \not \leq 5$$. These suggest the function $$g(b) = \lfloor b/2 \rfloor$$ is our maximum. More formally, we want $$g(b) = max$${$$a : 2a \leq b, a \in Z$$}…
• Puzzle 10 not decreasing sequences, and functions are monotonic. But this is sort of not in the spirit of the posets we're exploring. Puzzle 11 post fix If $$f : A \rightarrow B$$ and its inverse $$g: B \rightarrow A$$ are both monotonic. Then \…
• Puzzle 7. Let $$\leq$$ be a reflexive and transitive relation. The pairs in our relation are morphisms of a category. The Objects are the elements of the underlying set. The composition is defined as $$(x, y) \circ (y, z) = (x, z)$$ for all $$(x,y)\… • Real numbers between 0 and 1 are a poset. The map from events to their probabilities is then Monotonic. \(A \leq B \Rightarrow P(A) \leq P(B)$$