Problem 2 - Applied Discrete Structures

Question

Let \(A=\\{1,2,3\\} .\) Define \(f: A \rightarrow A\) by \(f(1)=2, f(2)=1,\) and \(f(3)=3\). Find \(f^{2}, f^{3}, f^{4}\) and \(f^{-1}\).

Step-by-Step Solution

Answer

\(f^2(x) = x\), \(f^3(x) = f(x)\), \(f^4(x) = x\), and \(f^{-1} = f\).

Step 1: Understand the function

The function \(f\) is defined from the set \(A = \{1, 2, 3\}\) to itself. It is given by the mappings: \(f(1) = 2\), \(f(2) = 1\), and \(f(3) = 3\). This function rearranges the elements of \(A\).

Step 2: Compute \(f^2\)

The square of the function, \(f^2(x) = f(f(x))\), means we apply \(f\) twice. Applying \(f\) to each element, \(f^2(1) = f(f(1)) = f(2) = 1\), \(f^2(2) = f(f(2)) = f(1) = 2\), and \(f^2(3) = f(f(3)) = f(3) = 3\). Therefore, \(f^2 = \{(1,1), (2,2), (3,3)\}\).

Step 3: Compute \(f^3\)

To find \(f^3(x) = f(f^2(x))\), apply the function three times. We've already determined \(f^2\). Thus, \(f^3(1) = f(f^2(1)) = f(1) = 2\), \(f^3(2) = f(f^2(2)) = f(2) = 1\), and \(f^3(3) = f(f^2(3)) = f(3) = 3\). Hence, \(f^3 = \{(1,2), (2,1), (3,3)\}\), which equals \(f\).

Step 4: Compute \(f^4\)

For \(f^4(x) = f(f^3(x))\), apply the function four times. Since \(f^3 = f\), we have \(f^4(1) = f^2(1) = 1\), \(f^4(2) = f^2(2) = 2\), and \(f^4(3) = f^2(3) = 3\). Therefore, \(f^4 = \{(1,1), (2,2), (3,3)\}\), which equals \(f^2\).

Step 5: Find the inverse function \(f^{-1}\)

The inverse function \(f^{-1}\) reverses the roles of inputs and outputs. Given \(f(1) = 2\), we have \(f^{-1}(2) = 1\); for \(f(2) = 1\), it follows that \(f^{-1}(1) = 2\); and for \(f(3) = 3\), \(f^{-1}(3) = 3\). Therefore, \(f^{-1} = \{(2,1), (1,2), (3,3)\}\).

Key Concepts

Inverse Function

An inverse function, denoted as \(f^{-1}\), is essentially a function that reverses the effects of the original function \(f\). To understand this better, imagine a simple action: opening a lock with a specific key. The function \(f\) represents using this key to unlock, while \(f^{-1}\) represents locking it back.For the function \(f\) given in our exercise, we defined it by the mappings \(f(1) = 2\), \(f(2) = 1\), and \(f(3) = 3\). The inverse function \(f^{-1}\) reverses these mappings. Hence, if \(f(1) = 2\), then \(f^{-1}(2) = 1\), and so on. This gives us the pairs \(f^{-1} = \{(2,1), (1,2), (3,3)\}\).Finding an inverse is crucial as it allows us to "undo" a function, converting outputs back to their original inputs.

Permutation Function

A permutation function rearranges the elements of a set in a specific order. In mathematics, particularly in discrete mathematics, a permutation is an arrangement of elements in a particular sequence or order.In our exercise, \(f\) is a permutation function on the set \(A = \{1, 2, 3\}\). This means it shuffles the elements of set \(A\), mapping the elements as \(f(1) = 2\), \(f(2) = 1\), and \(f(3) = 3\). Essentially, each element from set \(A\) is reordered according to the function \(f\).The interesting aspect of permutation functions is that repeated applications can yield the identity function, where each element maps onto itself again. Notice how \(f^2 = \{(1,1), (2,2), (3,3)\}\) is the identity permutation, showing all elements are mapped back to themselves.

Discrete Mathematics

Discrete mathematics involves studying mathematical structures that are fundamentally discrete, as opposed to continuous like real numbers. It's about distinct and separate values and includes topics such as graphs, integers, and, as used in this exercise, permutations and functions.Functions in discrete mathematics, like the function \(f\) from our exercise, operate on finite sets with distinct elements. Here, we explored how functions (like \(f\)) and operations (like composition resulting in \(f^2, f^3\)) are used to understand mappings and arrangements.Key concepts in discrete mathematics related to this exercise include:

• Functions and their compositions
• Inverse functions
• Permutation of finite sets

These ideas are critical in computer science, cryptography, and combinatorics, where discrete structures are often used to model and solve problems.