Question

Define permutations. Find the permutations of order 3 .

Short answer

A permutation is an arrangement of objects in a specific order, where each object appears exactly once. For a set of three distinct elements \(\{a, b, c\}\), the permutations of order 3 are: \(abc\), \(acb\), \(bac\), \(bca\), \(cab\), and \(cba\).

Step-by-step solution

Definition of Permutations

A permutation is an arrangement of objects in a specific order. In a permutation, each object appears exactly once, and the order in which the objects are arranged matters. More formally, a permutation of the set \(A\) is a bijection, which is a one-to-one correspondence (mapping) from the set \(A\) to itself. The set \(A\) can be finite or infinite.

Permutations of Order 3

In this context, "order 3" means that we are considering the permutations involving three distinct elements. Let's consider a set of three distinct elements: \(\{a, b, c\}\). The aim is to find all possible permutations of these three elements. To do this, we can follow these steps: 1. Select an element to be the first element of the permutation. 2. Select an element from the remaining elements to be the second element of the permutation. 3. Place the last remaining element as the third element of the permutation. Let's go through all the possible permutations step by step: 1. Start with the first element as \(a\): - Second element can be either \(b\) or \(c\): - If second element is \(b\), the third element is \(c\). So, the permutation is: \(abc\). - If second element is \(c\), the third element is \(b\). So, the permutation is: \(acb\). 2. Next, consider the first element as \(b\): - Second element can be either \(a\) or \(c\): - If the second element is \(a\), the third element is \(c\). So, the permutation is: \(bac\). - If the second element is \(c\), the third element is \(a\). So, the permutation is: \(bca\). 3. Finally, let the first element be \(c\): - Second element can be either \(a\) or \(b\): - If the second element is \(a\), the third element is \(b\). So, the permutation is: \(cab\). - If the second element is \(b\), the third element is \(a\). So, the permutation is: \(cba\). So, the permutations of order 3 for the set \(\{a,b,c\}\) are: \(abc\), \(acb\), \(bac\), \(bca\), \(cab\), and \(cba\).

Key concepts

Finite Sets

A finite set is a collection of distinct objects or elements that contains a limited number of items. Finite sets are important in mathematics because they allow us to count, measure, and organize data in a manageable way. For example:

• Think of the set \(\{1, 2, 3\}\), which contains exactly three elements. This is a finite set because you can list all the items.
• Another example might be the set of all students in a classroom. It has a definite number.

The concept of permutation is often explored using finite sets. This is because the arrangements of these elements are easier to handle when there's a clear limit, unlike with infinite sets where arrangements become unmanageable. A finite set allows us to examine each possible permutation clearly without becoming overwhelmed.

Bijection

A bijection is a one-to-one correspondence between two sets, where every element in one set maps to exactly one element in another set, and vice versa. This means that there are no leftovers or repeats. Here’s why it’s essential in understanding permutations:

• For a permutation of a set, you can consider it as a bijection from the set to itself.
• Imagine you have set \(A = \{a, b, c\}\). A bijection ensures that each element maps to a unique position—essential for forming valid permutations.

Bijection helps establish that each object can be moved to another position without violating the rule of uniqueness. In essence, bijections confirm that permutations are both injective (one-to-one) and surjective (onto), making them an exact reordering of the set elements.

Arrangement of Objects

The arrangement of objects refers to the specific order in which elements from a set are organized. In permutations, the sequence is key because changing the order results in a different permutation.

• Consider the set \(\{a, b, c\}\). Arranging these objects gives us different outcomes like \(abc\), \(acb\), \(bac\), and so forth.
• Each unique sequence indicates a different permutation, highlighting the importance of order.

Understanding how objects can be shuffled around the set not only assists in grasping permutations but also strengthens ideas about combinations and probability. Once you know the rules for arranging a small set, larger sets become simpler to handle, and this foundational idea can be applied to problems in various domains like algorithms and data understanding.