It's really exciting to see you here! Your answer on stack overflow made me aware of Christopher Strachey's denotational semantics. I grabbed Stoy's book, and from this, learned it's okay to think algebraically, before algorithmically. Thank you for…
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 \(g(b) = max\){\( a : 2a \leq b, a \in Z \)}…
On my phone it doesn't seem to display the category filters on the left side of the screen by default. You have to scroll to the bottom and press the "full site" link.
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)\…