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Exercises

1. [02] Consider the transformation of {0, 1, 2, 3, 4, 5, 6} that replaces x by 2x mod 7. Show that this transformation is a permutation, and write it in cycle form.

2. [10] The text shows how we might set (a, b, c, d, e, f) ← (c, d, f, b, e, a) by using a series of replacement operations (xy) and one auxiliary variable t. Show how to do the job by using a series of exchange operations (x ↔ y) and no auxiliary variables.

3. [03] Compute the product , and express the answer in two-line notation. (Compare with Eq. (4).)

4. [10] Express (a b d)(e f) (a c f) (b d) as a product of disjoint cycles.

5. [M10] Equation (3) shows several ...

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