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Chapter 2: Problem 44

Given the set \(\mathrm{S}=\\{1,2,3,4,5,6,\),\(} find a partition of \mathrm{S}\).

Short Answer

Expert verified
A possible partition of the set \(S = \{1,2,3,4,5,6\} \) can be divided based on even and odd numbers: - Subset A: \(\{1, 3, 5\} \) (odd numbers) - Subset B: \(\{2, 4, 6\} \) (even numbers)

Step by step solution

01

Understand the concept of a partition

A partition of S is a collection of non-empty, disjoint subsets of S, such that their union is S. In other words, each element of S must belong to one and only one of the subsets in the partition. We have to divide S into a finite number of subsets such that these two conditions are met.
02

Create a partition

To create a partition, we must divide the elements of S into non-empty, disjoint subsets. Since S has 6 elements, we have several possible ways to create a partition. Some possible partitions are: Option 1: Divide S based on even and odd numbers - Subset A: {1, 3, 5} (odd numbers) - Subset B: {2, 4, 6} (even numbers) Option 2: Divide S into two-element subsets - Subset A: {1,2} - Subset B: {3,4} - Subset C: {5,6} Option 3: Divide S into single-element subsets - Subset A: {1} - Subset B: {2} - Subset C: {3} - Subset D: {4} - Subset E: {5} - Subset F: {6} Each of these partitions divides S into disjoint subsets, and the union of the subsets in each option gives us the original set S.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Finite Sets
Finite sets are collections of distinct elements that have a countable number of members. In simple terms, you can count the elements without going on indefinitely. For example, the set \( S = \{1, 2, 3, 4, 5, 6\} \) is a finite set because it contains a specific number of elements, in this case, six.
When dealing with finite sets, you can easily enumerate or list all the elements. This characteristic makes operations like finding subsets, intersections, and unions easier to manage and understand.
  • Finite sets are often represented with curly braces \( \{ \} \) enclosing their elements.
  • The number of elements in a finite set is known as the cardinality. For our set \( S \), the cardinality is 6.
Understanding finite sets is crucial when working on problems involving partitions, as the limited number of elements simplifies the task of dividing the set into smaller sets.
Disjoint Subsets
Disjoint subsets are an essential concept in set theory, especially when discussing partitions. Two or more subsets are disjoint if they have no elements in common. In other words, the intersection of disjoint subsets is always an empty set \( \emptyset \).
When creating a partition of a finite set like \( S \), each subset in the partition must be disjoint from the others.
  • For example, in the partition into odd and even numbers, Subset A \( \{1, 3, 5\} \) and Subset B \( \{2, 4, 6\} \) are disjoint since they do not share any elements.
  • This characteristic helps ensure each element of the original set belongs only to one subset when forming a partition.
Disjoint subsets ensure clear and precise partitioning in set operations, avoiding any overlaps and maintaining the integrity of set division.
Set Theory Concepts
Set theory provides a foundational framework for understanding collections of objects, whether numbers, letters, or more abstract elements. Several basic concepts within set theory are vital for comprehending set partitions.
  • Union: The union of two sets contains all elements from both sets. For a partition, the union of all disjoint subsets must equal the original set.
  • Intersection: The intersection of sets includes only the elements common to all involved sets. For disjoint subsets, this is always an empty set.
  • Subset: A subset is a set where all its elements are also in another set. For instance, each of the parts of the partition, such as \( \{1, 2\} \) or \( \{3, 4\} \), is a subset of \( S \).
In the context of the given problem, understanding these set theory concepts helps you properly manipulate the elements to form accurate partitions.

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Most popular questions from this chapter

Prove that the empty set \(\varphi\) is a subset of every set.

Given: \(U=\\{a, b, c, 1,2,3\\}, A=\\{a, b, 3\\}\) and \(B=\\{a, 1,2,3\\}\) find: \(\quad\) (1) \(\mathrm{A}^{\prime} \cup \mathrm{B}^{\prime} \quad\) (2) \(\mathrm{A} \cap \mathrm{B}\) (3) \(\left(\mathrm{A} \cap \mathrm{B}^{\prime}\right) \cup\left(\mathrm{A}^{\prime} \cap \mathrm{B}\right)\)

Given the intervals \(\mathrm{A}=\\{\mathrm{a} \mid 0 \leq \mathrm{a} \leq 4\\}, \mathrm{B}=\\{\mathrm{b} \mid 1 \leq \mathrm{b} \leq 2\\}\), and \(\mathrm{C}=\\{\mathrm{c} \mid-1<\mathrm{c}<2\\}\) find \(\mathrm{A} \times \mathrm{B}\) and \(\mathrm{B} \times \mathrm{C}\). Also graph \(\mathrm{A} \times \mathrm{C}\) and \((-\infty, 0] \times[0, \infty)\).

Find the cardinal sum \(4+5\) of the two finite cardinal numbers 4 and 5 .

If \(\mathrm{A}=\\{(\mathrm{x}, \mathrm{y})|| \mathrm{x}-4 \mid<1\) and \(|\mathrm{y}-3|<1\\}\) and \(B=\left\\{(x, y) \mid x^{2}+y^{2}-8 x-6 y+21 \leq 0\right\\} .\) Show that \(A \subset B\)
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