In this problem, we shall use four transformations, I, R, S and T.
Their effects are shown below.
We write R-1 for the transformation that ``undoes'' R (the inverse of
R), and R S for "do R, then S".
We can write T followed by T as T T or T2, and T followed by T
followed by T as T T T or T3 and so on.
Similarly, we
can write S-1S-1 as S-2 and so on.
Try to find simpler ways to write: R2, R3, R4, ... S2, S3, S4, ... T2, T3, T4, ....
What do you notice?
Can you find a simpler way to write R2006 and
S2006?
Can you describe T2006?
Let's think about the order in which we carry out transformations:
What
happens if you do R S? Do you think that S R
will be the same? Try it and see.
Is T2R the same as R T2?
Is (R T)S the same as S(R T)?
Try this with some other transformations. Does changing the order
always
sometimes
never
produce the same transformation?
Now let's think about how to undo R S. What combination of I, R, S,
T and their inverses might work? Try it and see: does it work? If not,
why not? Can you find
a combination of transformations that does work?
How can you undo
transformations like S T, T R and R S2?
