16.1.1 Permutation of a Set A

#Type/Definition #Topic/Linear_Algebra #Type/Example

Types: N/A
Examples: N/A
Constructions: 16.1.2 The Set of All Permutations of n Elements
Generalizations: 16.1 Permutation

Properties: N/A
Sufficiencies: N/A
Questions: N/A

Permutation of a Set

A

Let $A$ be a nonempty set. A permutation of A is a bijective function from $A$ to itself. In other words, a 16.1 Permutation is a map

π : A \to A

such that

$π$ is one-to-one: if $π (a) = π (b)$ , then $a = b$
$π$ is onto: $Codomain (π) = Rng (π)$

Remark. Permutations are the tool used when rearranging the order of a specific set.
$⧫$

Tire Change

Imagine we want to rotate the tires of a car during a tire change. At the start of the job, each of the four tires is labeled in one of four positions. We define the set $A = {1, 2, 3, 4}$ used to label each tire location and assign each tire in that location a number in $A$ .

To rotate the tires, we want to place each tire in one of the four positions on the car. No two tires can be in one position, and the tires should be rearranged so no tire is not in the same place it was previously. One possible rearrangement is to let 16.1 Permutation $f : A \to A$ on the set $A = {1, 2, 3, 4}$ .

\begin{aligned} f (1) = 4, & f (2) = 3, & f (3) = 2, & f (4) = 1 \end{aligned}

This permutation would swap the top left and bottom right and swap the front right and back left.

Our mapping $f$ is injective because each $i \in A$ has a unique image under $f$ . We also see that $f$ is surjective since $Rng (f) = A$ .