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Chapter 1: Problem 83

List all partitions of the set. $$ \\{1\\} $$

Short Answer

Expert verified
The only partition of the set \{1\} is \{ \{1\} \}.

Step by step solution

01

Understand the Concept of Partitioning

Before getting started, get a clear understanding of the term 'partitioning'. A partition of a set is a grouping of the set's elements into non-empty subsets, in such a way that every element is included in one and only one of the subsets.
02

Identify the Elements of the Set to Partition

The given set is \{1\}. It only contains a single element, '1'.
03

List All Partitions

Since there's only one element in this set, there's only one way to partition it: \{ \{1\} \}. The set itself is the only partition. We cannot create other partitions without violating the rules of partitioning.

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Key Concepts

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

Partitions of a Set
Set partitioning is a fundamental concept in mathematics, especially in combinatorics. When we talk about partitions of a set, we are referring to a way of splitting all its elements into distinct, non-overlapping groups. Each element must belong to exactly one subgroup. This ensures that the original set is fully encompassed by these groups with no elements left out or repeated elsewhere in another group.

A partition must adhere to these rules:
  • Each subgroup (or subset) must not be empty.
  • All elements from the original set need to be included in some subset.
  • Subsets must be distinct and non-overlapping, meaning an element cannot appear in more than one subset.
These fundamental rules ensure that the way we group elements is systematic and thorough, creating valid partitions of the original set.
Single Element Set Partition
When considering the set \(\{1\}\), which contains just a single element, the concept of partitioning becomes very straightforward. In this particular case, creating a partition is simple because there's only one way to group the elements.

Partitioning a single element set, such as \(\{1\}\), inevitably leads to the formation of only one subset, which is the set itself \(\{\{1\}\}\). This is because:
  • There are no other elements to group with '1', so '1' must remain in its own separate subset.
  • Attempting to form additional subsets would either leave some elements out or result in duplication, both of which violate the rules of partitioning.
Thus, whenever dealing with a single element set, its partition is self-evident and consists of the element alone.
Non-Empty Subsets
In the context of partitions, subsets formed during a partition must be non-empty. This means that when partitioning a set, every subset that is created must contain at least one of the original set's elements.

Considerations when forming non-empty subsets involve:
  • Ensuring that each subset has elements; a subset with no elements (an empty set) does not count as a valid partition.
  • Subsets collectively should include all elements from the original set, emphasizing completeness and coverage.
In our example of \(\{1\}\), the subset \(\{\{1\}\}\) is non-empty as it includes the single element '1'. Non-empty subsets in partitions guarantee that no portion of the original set is neglected, maintaining the integrity of the partitioning exercise.

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Most popular questions from this chapter

Give an argument using rules of inference to show that the conclusion follows from the hypotheses. Hypotheses: Everyone in the class has a graphing calculator. Everyone who has a graphing calculator understands the trigonometric functions. Conclusion: Ralphie, who is in the class, understands the trigonometric functions.

For each pair of propositions \(P\) and \(Q\) . State whether or not \(P \equiv Q\). $$ P=p \rightarrow q, Q=\neg p \vee q $$

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Answer true or false. $$ \\{x\\} \subseteq\\{x\\} $$

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