Question

Find the inverse of each of the following permutations: \(\pi_{1}=\) \(\left(\begin{array}{llll}1 & 2 & 3 & 4 \\ 3 & 2 & 4 & 1\end{array}\right), \pi_{2}=\left(\begin{array}{llll}1 & 2 & 3 & 4 & 5 \\ 1 & 4 & 2 & 5 & 3\end{array}\right), \pi_{3}=\left(\begin{array}{llll}1 & 2 & 3 & 4 & 5 & 6 \\\ 6 & 5 & 4 & 3 & 2 & 1\end{array}\right)\), and \(\pi_{4}=\left(\begin{array}{llll}1 & 2 & 3 & 4 & 5 \\ 3 & 4 & 5 & 1 & 2\end{array}\right) .\)

Short answer

The inverse of the permutations are \(\pi_{1}^{-1}=\left(\begin{array}{llll}3 & 2 & 4 & 1\1 & 2 & 3 & 4\end{array}\right)\), \(\pi_{2}^{-1}=\left(\begin{array}{lllll}1 & 4 & 2 & 5 & 3\1 & 2 & 3 & 4 & 5\end{array}\right)\), \(\pi_{3}^{-1}=\left(\begin{array}{llllll}6 & 5 & 4 & 3 & 2 & 1\1 & 2 & 3 & 4 & 5 & 6\end{array}\right)\) and \(\pi_{4}^{-1}=\left(\begin{array}{lllll}3 & 4 & 5 & 1 & 2\1 & 2 & 3 & 4 & 5\end{array}\right)\)

Step-by-step solution

Find the inverse of \(\pi_{1}\)

Exchange the first and second rows to inverse \(\pi_{1}\). The inverse of the permutation, denoted as \(\pi_{1}^{-1}\), is as follows, \(\pi_{1}^{-1}=\left(\begin{array}{llll}3 & 2 & 4 & 1\1 & 2 & 3 & 4\end{array}\right)\)

Find the inverse of \(\pi_{2}\)

Exchange the first and second rows to find the inverse of \(\pi_{2}\). The inverse of the permutation, denoted as \(\pi_{2}^{-1}\), is as follows, \(\pi_{2}^{-1}=\left(\begin{array}{lllll}1 & 4 & 2 & 5 & 3\1 & 2 & 3 & 4 & 5\end{array}\right)\)

Find the inverse of \(\pi_{3}\)

Exchange the first and second rows to find the inverse of \(\pi_{3}\). The inverse of the permutation, denoted as \(\pi_{3}^{-1}\), is as follows, \(\pi_{3}^{-1}=\left(\begin{array}{llllll}6 & 5 & 4 & 3 & 2 & 1\1 & 2 & 3 & 4 & 5 & 6\end{array}\right)\)

Find the inverse of \(\pi_{4}\)

Exchange the first and second rows to find the inverse of \(\pi_{4}\). The inverse of the permutation, denoted as \(\pi_{4}^{-1}\), is as follows, \(\pi_{4}^{-1}=\left(\begin{array}{lllll}3 & 4 & 5 & 1 & 2\1 & 2 & 3 & 4 & 5\end{array}\right)

Key concepts

Permutation Notation

In mathematics, permutation notation is a way to represent how elements of a set are rearranged or permuted. Imagine a set of items, like numbers, and consider how you might change their order. This rearrangement is captured through permutation notation. Permutations are typically represented using a two-row symbol where the top row contains the elements in their original order, and the bottom row shows their new position after the permutation.

For example, consider the permutation \( \pi_1 \) given in the exercise: \[ \pi_{1}= \begin{pmatrix} 1 & 2 & 3 & 4 \ 3 & 2 & 4 & 1 \end{pmatrix} \] This means that the number 1 is moved to the position of 3, 2 stays as 2, 3 goes to the position of 4, and 4 moves to 1's position.

• Top row: Original positions.
• Bottom row: New positions.

Understanding permutations in this way is crucial for grasping other mathematical concepts, such as substitution in algebra or higher-level topics like group theory.

Inverse Functions

The concept of inverse functions is integral to understanding how relationships between mathematical entities can be reversed. If you think of a function as a process or rule that transforms an input into an output, an inverse function reverses this process. It takes the output back to the original input.

In the context of permutation notation, finding the inverse of a permutation involves determining how to return elements to their original positions. Essentially, you swap the two rows of the permutation notation. Let's revisit the permutation \( \pi_1 \) and find its inverse: \[ \pi_{1}^{-1} = \begin{pmatrix} 3 & 2 & 4 & 1 \ 1 & 2 & 3 & 4 \end{pmatrix} \] Here, the swapping of the rows indicates how each element is returned to its original location. This concept of finding inverses is widespread in mathematics, such as in solving equations or determining inverse matrices.

Key aspects of inverse functions include:

• Reversing a specific operation or process.
• Returning elements to their initial state.
• Maintaining the properties and relationships governed by the original function.

Understanding inverse functions helps build a fundamental understanding of modular operations and reversibility within mathematics.

Mathematics Education

Teaching mathematics and, more specifically, concepts like permutations and inverse functions, involves breaking down complex ideas into simpler, more digestible parts. Understanding and visualizing permutations helps students create mental models that aid in solving more complex problems.

Effective mathematics education embraces multiple strategies, such as:

• Using visual aids to illustrate how elements are swapped in a permutation.
• Gradually introducing the concept of inverses, first in simple processes before moving to more abstract ideas like function inverses.
• Incorporating exercises and real-world examples to contextualize these concepts.

By focusing on the simplified concept of swapping rows in permutation notation, educators can forge connections to broader mathematical ideas. This methodology not only supports mathematics comprehension but also encourages critical thinking and problem-solving skills, equipping students with the tools they need for diverse mathematical challenges. Understanding permutations and their inverses lays important groundwork for higher-level math concepts, including algebra and calculus.