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Chapter 9: Problem 42

Which of these collections of subsets are partitions of \(\\{-3,-2,-1,0,1,2,3\\} ?\) a) \(\\{-3,-1,1,3\\},\\{-2,0,2\\}\) b) \(\\{-3,-2,-1,0\\},\\{0,1,2,3\\}\) c) \(\\{-3,3\\},\\{-2,2\\},\\{-1,1\\},\\{0\\}\) d) \(\\{-3,-2,2,3\\},\\{-1,1\\}\)

Short Answer

Expert verified
(a) and (c) are partitions.

Step by step solution

01

Define a Partition

A partition of a set is a way of dividing the set into non-empty subsets such that every element of the set is included in exactly one subset.
02

Check Collection (a)

Examine if \(\{-3,-1,1,3\},\{-2,0,2\}\) covers all elements of the set \({-3,-2,-1,0,1,2,3}\). Each element is included, and no element is repeated. Thus, collection (a) is a partition.
03

Check Collection (b)

Examine if \(\{-3,-2,-1,0\},\{0,1,2,3\}\) covers all elements of the set. Notice that 0 appears in both subsets, which violates the partition rule. Therefore, collection (b) is not a partition.
04

Check Collection (c)

Examine if \(\{-3,3\},\{-2,2\},\{-1,1\},\{0\}\) covers all elements of the set. Each subset contains unique elements and all elements of the set are included exactly once. Thus, collection (c) is a partition.
05

Check Collection (d)

Examine if \(\{-3,-2,2,3\},\{-1,1\}\) covers all elements of the set. Notice that elements 0 and 2 are repeated and not all elements are covered. Therefore, collection (d) is not a partition.

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Key Concepts

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

Discrete Mathematics
Discrete Mathematics is a branch of mathematics that deals with distinct and separate elements. Unlike continuous mathematics, which deals with values that vary smoothly, discrete mathematics deals with objects that can assume only distinct, separated values. It includes a variety of topics that are essential in computer science and other applications.

Some key concepts in discrete mathematics include:
  • Graphs and Trees
  • Combinatorics
  • Logic and Set Theory
  • Number Theory
Understanding discrete mathematics provides a strong foundation for problem-solving and algorithm development. It is particularly useful in the context of computer science for tasks such as database management, cryptography, and network algorithms.
Set Theory
Set theory is a fundamental part of modern mathematics, dealing with the study of collections of objects, which are called sets. It forms the basis for various other fields of mathematics. A set is simply a collection of distinct objects, considered as an object in its own right. An element is any one of the objects in the set.

Some basic operations and concepts in set theory include:
  • Union: Combining all elements from two sets.
  • Intersection: The set of elements common to both sets.
  • Difference: Elements in one set but not in another.
  • Subset: A set where all its elements are contained within another set.
In the context of the exercise, we are concerned with partitions of sets. A partition divides a set into non-overlapping subsets, such that every element is included in exactly one subset. This idea is crucial for various applications including data segmentation and analysis.
Subsets
A subset is a portion of a set, where every element of the subset is also an element of the original set. In formal terms, a set A is a subset of a set B if every element of A is also an element of B. This is denoted as \( A \subseteq B \).

Subsets can be proper or improper:
  • Proper Subsets: Every element of A is in B, and B has at least one element not in A.
  • Improper Subsets: A and B are exactly the same set, where every element of A is in B and vice versa.
To understand the exercise fully, it is important to grasp that a partition involves dividing a set into several non-empty subsets such that each element of the set is included in one and only one subset. For example, in the solution provided, examining the sets for coverage and repetition is key to determining if they form a valid partition.

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Most popular questions from this chapter

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.

Find the smallest equivalence relation on the set \(\\{a, b, c, d, e\\}\) containing the relation \(\\{(a, b),(a, c),(d, e)\\}\)

a) Show that the least upper bound of a set in a poset is unique if it exists. b) Show that the greatest lower bound of a set in a poset is unique if it exists.

Show that the poset of rational numbers with the usual less than or equal to relation, \((\mathbf{Q}, \leq),\) is a dense poset.

Suppose that \(R_{1}\) and \(R_{2}\) are equivalence relations on the set \(S .\) Determine whether each of these combinations of \(R_{1}\) and \(R_{2}\) must be an equivalence relation. $$ \begin{array}{llll}{\text { a) } R_{1} \cup R_{2}} & {\text { b) } R_{1} \cap R_{2}} & {\text { c) } R_{1} \oplus R_{2}}\end{array} $$
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