Remark 21.2.1.
Once you have written the elements of a finite set in some order, think of a permutation as a way of re-ordering them.
Section 21.2 Definition
We often want to count how many ways we can “mix up” the objects in a collection.
- permutation
- a bijection from a finite set to itself
Example 21.2.2. Permutations of three objects.
Figure 21.2.3 contains tables of values for all six possible permutations of the set \(A = \{a,b,c\}\text{.}\) We have grouped them according to: all elements fixed; one element fixed and two mixed; all elements mixed.
| \(x\) | \(a\) | \(b\) | \(c\) |
| \(\id_A(x)\) | \(a\) | \(b\) | \(c\) |
| \(x\) | \(a\) | \(b\) | \(c\) |
| \(f_a(x)\) | \(a\) | \(c\) | \(b\) |
| \(f_b(x)\) | \(c\) | \(b\) | \(a\) |
| \(f_c(x)\) | \(b\) | \(a\) | \(c\) |
| \(x\) | \(a\) | \(b\) | \(c\) |
| \(s_r(x)\) | \(c\) | \(a\) | \(b\) |
| \(s_l(x)\) | \(b\) | \(c\) | \(a\) |
