Let A = {1, 2, 3, 6, 9, 18} and define R on A by xRy if x|y. Draw the Hasse diagram for the poset (A,R).
How would I go about doing this? Thanks! :)
How would I go about doing this? Thanks! :)
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Put 18 on the top line of a paper.
Put 6 and 9 below that on a second line.
But 2 and 3 on a line below that.
But 1 on a line below that.
When I say put these things "on a line" I just mean align them horizontally so they are next to each other. I don't actually mean draw line segments between them or anything.
Now that you've got that, draw a line from 1 to 2 and from 1 to 3.
Draw another line from 2 to 6 and from 3 to 6 and from 3 to 9.
Draw another line from 6 to 18 and from 9 to 18.
Now you're done.
The point of the Hasse diagram is that there should be a line from a to b if a | b and if there is no number c that isn't equal to a or b with a | c and c | b. When you complete the picture you can see that if a and b are elements of A, then a divides b if and only if either a = b or there is a chain of line segments from a up to b. There is a line segment from a to b precisely when a divides b and there is no number c in A unequal to a or b with the property that a | c and c | b. The reason you don't put a line segment from 1 to 18 (for example) is that although 1 divides 18, the fact that 1 divides 18 is easy to see from the diagam (e.g. because the lines already drawn tell you that 1 divides 2, and 2 divides 6, and 6 divides 18, and you know 'divides' is a transitive relation) so there is no reason for a line to convey this information. The condition for when you draw lines in a Hasse diagram ensures that you put a line from a to b only when it is "absolutely necessary" and not a consequence of other lines you have drawn. I am speaking a little informally here but hopefully it helps to convey the idea.
Put 6 and 9 below that on a second line.
But 2 and 3 on a line below that.
But 1 on a line below that.
When I say put these things "on a line" I just mean align them horizontally so they are next to each other. I don't actually mean draw line segments between them or anything.
Now that you've got that, draw a line from 1 to 2 and from 1 to 3.
Draw another line from 2 to 6 and from 3 to 6 and from 3 to 9.
Draw another line from 6 to 18 and from 9 to 18.
Now you're done.
The point of the Hasse diagram is that there should be a line from a to b if a | b and if there is no number c that isn't equal to a or b with a | c and c | b. When you complete the picture you can see that if a and b are elements of A, then a divides b if and only if either a = b or there is a chain of line segments from a up to b. There is a line segment from a to b precisely when a divides b and there is no number c in A unequal to a or b with the property that a | c and c | b. The reason you don't put a line segment from 1 to 18 (for example) is that although 1 divides 18, the fact that 1 divides 18 is easy to see from the diagam (e.g. because the lines already drawn tell you that 1 divides 2, and 2 divides 6, and 6 divides 18, and you know 'divides' is a transitive relation) so there is no reason for a line to convey this information. The condition for when you draw lines in a Hasse diagram ensures that you put a line from a to b only when it is "absolutely necessary" and not a consequence of other lines you have drawn. I am speaking a little informally here but hopefully it helps to convey the idea.