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Chapter 9: Problem 41

Which of these collections of subsets are partitions of \(\\{1,2,3,4,5,6\\} ?\) $$ \begin{array}{l}{\text { a) }\\{1,2\\},\\{2,3,4\\},\\{4,5,6\\}} \\ {\text { b) }\\{1\\},\\{2,3,6\\},\\{4\\},\\{5\\}} \\ {\text { c) }\\{2,4,6\\},\\{1,3,5\\}} & {\text { d) }\\{1,4,5\\},\\{2,6\\}}\end{array} $$

Short Answer

Expert verified
Options b and c are partitions.

Step by step solution

01

Understanding Partitions

A partition of a set is a collection of non-empty subsets such that every element is included in exactly one subset. Therefore, the union of all subsets must be the original set, and subsets must be mutually exclusive.
02

Check Option a

Consider the sets {1,2}, {2,3,4}, {4,5,6}. Check for union and mutual exclusivity:1. Union: {1,2} 鈭� {2,3,4} 鈭� {4,5,6} = {1,2,3,4,5,6}2. Check mutual exclusivity: There is an overlap - 2 is in two subsets. Therefore, these sets are not mutually exclusive.
03

Check Option b

Consider the sets {1}, {2,3,6}, {4}, {5}. Check for union and mutual exclusivity:1. Union: {1} 鈭� {2,3,6} 鈭� {4} 鈭� {5} = {1,2,3,4,5,6}2. Check mutual exclusivity: No elements overlap. Therefore, all sets are mutually exclusive. This is a partition.
04

Check Option c

Consider the sets {2,4,6}, {1,3,5}. Check for union and mutual exclusivity:1. Union: {2,4,6} 鈭� {1,3,5} = {1,2,3,4,5,6}2. Check mutual exclusivity: No elements overlap. Therefore, these sets are mutually exclusive. This is a partition.
05

Check Option d

Consider the sets {1,4,5}, {2,6}. Check for union and mutual exclusivity:1. Union: {1,4,5} 鈭� {2,6} = {1,2,4,5,6}, but element 3 is missing.2. Therefore, it does not cover the whole set {1,2,3,4,5,6}. This is not a partition.

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Key Concepts

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

Mutually Exclusive Subsets
Mutually exclusive subsets are key for understanding partitions of a set. These subsets have no elements in common. If an element appears in one subset, it cannot appear in another.
This ensures that each element belongs to only one subset. Take option (a) from our exercise: these subsets are not mutually exclusive because element 2 appears in both {1,2} and {2,3,4}.
This overlap shows they aren't mutually exclusive. For a true partition, elements need to be in distinct subsets with no repetition.
Set Union
Understanding the union of sets helps in determining partitions. The union of sets combines all their elements without repetition.
For example, in option (c), sets {2,4,6} and {1,3,5} combine to form {1,2,3,4,5,6}. This covers all elements of the original set, showing completeness.

Union is essential because, in a valid partition, the union must cover the entire original set. If any element is missing after taking the union, like in option (d) where 3 is missing, then the collections of subsets cannot be considered a partition.
Discrete Mathematics
Discrete mathematics deals with distinct and separated values. Unlike continuous mathematics, which deals with smooth and unbroken lines, discrete math focuses on countable, separate elements.

In the context of our exercise, we're discussing how discrete elements (numbers) can be grouped into subsets. Each subset is distinct and shouldn't overlap with others when forming a partition.
It's this clear division and precise grouping that make discrete mathematics helpful for tasks like setting up partitions, ensuring there's no ambiguity in the categorization of elements.

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Most popular questions from this chapter

Show that the relation \(R=\emptyset\) on the empty set \(S=\emptyset\) is reflexive, symmetric, and transitive.

Show that the relation \(R\) on a set \(A\) is reflexive if and only if the complementary relation \(\overline{R}\) is irreflexive.

A partition \(P_{1}\) is called a refinement of the partition \(P_{2}\) if every set in \(P_{1}\) is a subset of one of the sets in \(P_{2}\) . Show that the partition of the set of people living in the United States consisting of subsets of people living in the same county (or parish) and same state is a refinement of the partition consisting of subsets of people living in the same state.

Which of these collections of subsets are partitions of the set of bit strings of length 8? a) the set of bit strings that begin with 1, the set of bit strings that begin with 00, and the set of bit strings that begin with 01 b) the set of bit strings that contain the string 00, the set of bit strings that contain the string 01, the set of bit strings that contain the string 10, and the set of bit strings that contain the string 11 c) the set of bit strings that end with 00, the set of bit strings that end with 01, the set of bit strings that end with 10, and the set of bit strings that end with 11 d) the set of bit strings that end with 111, the set of bit strings that end with 011, and the set of bit strings that end with 00 e) the set of bit strings that contain 3k ones for some nonnegative integer k, the set of bit strings that contain 3k + 1 ones for some nonnegative integer k, and the set of bit strings that contain 3k + 2 ones for some nonnegative integer k.

a) How many relations are there on the set \(\\{a, b, c, d\\} ?\) b) How many relations are there on the set \(\\{a, b, c, d\\}\) that contain the pair \((a, a) ?\)
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