(no subject)
sometimes i wish i could go from wake to sleep without that awkward phase of random thoughts.
it occurs to me that the speeds on my ceiling fan (off, fast, medium, slow) form a group over the operation "pull."
(if the fan is off, pulling once brings the speed to fast, a second time brings it to medium, a third pull makes it go slow, and a fourth pull returns it to its original off state.)
off pull off equals off (*)
off pull fast equals fast
off pull medium equals medium
off pull slow equals slow
fast pull off equals fast
fast pull fast equals medium
fast pull medium equals slow
fast pull slow equals off (*)
medium pull off equals medium
medium pull fast equals slow
medium pull medium equals off (*)
medium pull slow equals fast
slow pull off equals slow
slow pull fast equals off (*)
slow pull medium equals fast
slow pull slow equals medium
to be a group, the set of elements over the given operation must have:
closure: can be observed on the table.
associativity: the table only shows commutativity, but if you don't believe me and are willing to take the time, associativity can be proven.
identity element: off is the identity element, as proven by all equations that contain off.
inverse elements: the equations with a * after them indicate equations that verify the existance of inverse elements.
essentially, my ceiling fan is Z4 over addition. i just didn't want to say that initially because Z4 has nifty properties with other operations that my ceiling fan lacks.
it is also worth noting that:
the light attached to my ceiling fan is also a group, though that group only has two elements: "on" and "off."
a group is also formed by those lamps that have the little twisty thing to turn on, off, and to different levels of intensity.
my box fan is not a group, as it has two operations, "clockwise" and "counterclockwise" and can't be a ring because it has closure over neither operation.
my oscillating desk fan can't be a group as it has four operations and it doesn't have associativity over any of them.
a standard issue light switch is not a group for the same reasons as the box fan, it has too many operations and no closure.
*sigh* i'm so strange.
it occurs to me that the speeds on my ceiling fan (off, fast, medium, slow) form a group over the operation "pull."
(if the fan is off, pulling once brings the speed to fast, a second time brings it to medium, a third pull makes it go slow, and a fourth pull returns it to its original off state.)
off pull off equals off (*)
off pull fast equals fast
off pull medium equals medium
off pull slow equals slow
fast pull off equals fast
fast pull fast equals medium
fast pull medium equals slow
fast pull slow equals off (*)
medium pull off equals medium
medium pull fast equals slow
medium pull medium equals off (*)
medium pull slow equals fast
slow pull off equals slow
slow pull fast equals off (*)
slow pull medium equals fast
slow pull slow equals medium
to be a group, the set of elements over the given operation must have:
closure: can be observed on the table.
associativity: the table only shows commutativity, but if you don't believe me and are willing to take the time, associativity can be proven.
identity element: off is the identity element, as proven by all equations that contain off.
inverse elements: the equations with a * after them indicate equations that verify the existance of inverse elements.
essentially, my ceiling fan is Z4 over addition. i just didn't want to say that initially because Z4 has nifty properties with other operations that my ceiling fan lacks.
it is also worth noting that:
the light attached to my ceiling fan is also a group, though that group only has two elements: "on" and "off."
a group is also formed by those lamps that have the little twisty thing to turn on, off, and to different levels of intensity.
my box fan is not a group, as it has two operations, "clockwise" and "counterclockwise" and can't be a ring because it has closure over neither operation.
my oscillating desk fan can't be a group as it has four operations and it doesn't have associativity over any of them.
a standard issue light switch is not a group for the same reasons as the box fan, it has too many operations and no closure.
*sigh* i'm so strange.