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Draw the Hasse diagrams for \(P(\emptyset)\), \(P\{1\}\), and \(P\{1, 2\}\) where \(P(X)\) is the power set of \(X\), e.g.
$$ P\{1, 2\} = \{∅, \{1\}, \{2\}, \{1,2\}\}. $$
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Hello, I was doing this exercise. On page 13 of the book, there is a hint: "a powerset of a finite set, say P{0,1,2,...,n}, always looks like a cube of dimension n." Example 1.36 shows a cube of dim. 3 for P{0,1,2}.
In the exercise, I guess that the Hasse diagram of P{1,2} is a square -- e.g., as a cube can be seen in Flatland -- the Hasse diagram of P{1} is just an arrow between two points -- e.g., as a square can be seen in Lineland -- and the Hasse diagram of P{∅} is just the ∅ itself with no arrows -- e.g., as a line looks like in Pointland. The number of objects is a power of 2, where the exponent of 2 is the number of dimensions. The Hasse diagram of P{1,2}, if I did not make any mistakes, looks like a square with arrows from ∅ to {1}, from ∅ to {2}, from {1} to {1,2}, and from {2} to {1,2}. Please, let me know if you think this is correct. I'm new in this course, and it's nice to learn and share ideas.
Hello, I was doing this exercise. On page 13 of the book, there is a hint: "a powerset of a finite set, say P{0,1,2,...,n}, always looks like a cube of dimension n." Example 1.36 shows a cube of dim. 3 for P{0,1,2}. In the exercise, I guess that the Hasse diagram of P{1,2} is a square -- e.g., as a cube can be seen in Flatland -- the Hasse diagram of P{1} is just an arrow between two points -- e.g., as a square can be seen in Lineland -- and the Hasse diagram of P{∅} is just the ∅ itself with no arrows -- e.g., as a line looks like in Pointland. The number of objects is a power of 2, where the exponent of 2 is the number of dimensions. The Hasse diagram of P{1,2}, if I did not make any mistakes, looks like a square with arrows from ∅ to {1}, from ∅ to {2}, from {1} to {1,2}, and from {2} to {1,2}. Please, let me know if you think this is correct. I'm new in this course, and it's nice to learn and share ideas.That sounds about right! In fact, the Hasse diagram for \(\mathcal{P}\{0, 1, 2\}\) in the book contains all three of these. As you noticed, this is a cube -- and one of the faces of this cube is a square giving the Hasse diagram for \(\mathcal{P}\{1, 2\}\)! One of the edges on this face gives the Hasse diagram for \(\mathcal{P}\{1\}\), and the bottom-most vertex gives the Hasse diagram for \(\mathcal{P}\emptyset\).
That sounds about right! In fact, the Hasse diagram for \\(\mathcal{P}\\{0, 1, 2\\}\\) in the book contains all three of these. As you noticed, this is a cube -- and one of the faces of this cube is a square giving the Hasse diagram for \\(\mathcal{P}\\{1, 2\\}\\)! One of the edges on this face gives the Hasse diagram for \\(\mathcal{P}\\{1\\}\\), and the bottom-most vertex gives the Hasse diagram for \\(\mathcal{P}\emptyset\\).I think you're right, Maria! Here is a picture of \(P(\{0,1,2\})\):
and here is a picture of \(P(\{0,1,2,3\})\), drawn in a different style, where we write a bit to say whether each element is in or out of the subset:
You can see that this is a 4-dimensional cube.
I think you're right, Maria! Here is a picture of \\(P(\\{0,1,2\\})\\): <center><img src = "http://math.ucr.edu/home/baez/mathematical/7_sketches/P3_hasse_diagram.png"></center> and here is a picture of \\(P(\\{0,1,2,3\\})\\), drawn in a different style, where we write a bit to say whether each element is in or out of the subset: <center><img src = "http://math.ucr.edu/home/baez/mathematical/7_sketches/P4_hasse_diagram.png"></center> You can see that this is a 4-dimensional cube.Thank you! The 4-dimensional cube looks great. It's nice to see how geometry arises so naturally. Personally, I better understand math when I can come up with some visualization strategy.
Thank you! The 4-dimensional cube looks great. It's nice to see how geometry arises so naturally. Personally, I better understand math when I can come up with some visualization strategy.