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Chapter 9: Problem 17

Find the lexicographic ordering of these \(n\) -tuples: a) \((1,1,2),(1,2,1)\) b) \((0,1,2,3),(0,1,3,2)\) c) \((1,0,1,0,1),(0,1,1,1,0)\)

Short Answer

Expert verified
a) (1,1,2) < (1,2,1)b) (0,1,2,3) < (0,1,3,2)c) (0,1,1,1,0) < (1,0,1,0,1)

Step by step solution

01

- Compare the first elements

Begin by comparing the first elements of the tuples. The tuple with the smaller first element is lexicographically smaller. If they are equal, move to the next element.
02

- Apply to part (a)

Comparing the first elements of (1,1,2) and (1,2,1): both are 1, so move to the second element. Comparing the second elements: 1 < 2, so (1,1,2) < (1,2,1).
03

- Apply to part (b)

Comparing the first elements of (0,1,2,3) and (0,1,3,2): both are 0, so move to the second element. Comparing the second elements: both are 1, so move to the third element. Comparing the third elements: 2 < 3, so (0,1,2,3) < (0,1,3,2).
04

- Apply to part (c)

Comparing the first elements of (1,0,1,0,1) and (0,1,1,1,0): 0 < 1, so (0,1,1,1,0) < (1,0,1,0,1).

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Key Concepts

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

n-tuples
An n-tuple is a generalization of ordered pairs, triples, and so on. It is simply a collection of n elements in a specific order. For example, a 2-tuple is like \((1, 2)\), a 3-tuple is \((1, 2, 3)\), and so on. The number n represents the number of elements in the tuple.

Tuples are fundamental in mathematics because they help us work with ordered collections. Unlike sets, the order of elements matters in tuples. Thus, \((1, 2, 3)\) and \((3, 2, 1)\) are different tuples.

In the given exercise, we're dealing with tuples of different lengths (3-tuples and 5-tuples), which involve lists of numbers that must be considered in their specific order when determining their relationship to each other.
Comparison
Comparing tuples involves looking at their elements in order. Lexicographic ordering is a method used, which resembles the way words are ordered in a dictionary. Here鈥檚 how it works:

  • Start with the first element of each tuple.
  • If one number is smaller, that tuple is considered smaller overall.
  • If they are equal, move to the next element and repeat the process.
Let's apply this to one example from the exercise:

Part (a): Compare \((1, 1, 2)\) and \((1, 2, 1)\). First elements are the same (1 and 1). Compare the second elements: 1 < 2. Hence, \((1, 1, 2) < (1, 2, 1)\).

This step-by-step comparison ensures we precisely determine the lexicographic order.
Discrete Mathematics
Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. It includes topics such as logic, sets, relations, functions, and particularly, sequences and mappings.

Understanding tuples and their comparisons falls under discrete mathematics because tuples are ordered, finite sequences of elements. Discrete mathematics' principles and logic help us formalize and solve problems involving these structures.

In the context of our exercise, discrete mathematics provides the framework for:
  • Creating the tuples
  • Comparing their elements
  • Determining the order based on specific rules like lexicographic ordering


This structured approach makes it easier to handle complex problems where order and element comparisons are essential.

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Most popular questions from this chapter

Suppose that \(R\) and \(S\) are reflexive relations on a set \(A .\) Prove or disprove each of these statements. a) \(R \cup S\) is reflexive. b) \(R \cap S\) is reflexive. c) \(R \oplus S\) is irreflexive. d) \(R-S\) is irreflexive. e) \(S \circ R\) is reflexive.

Let \(R_{1}\) and \(R_{2}\) be relations on a set \(A\) represented by the matrices $$\mathbf{M}_{R_{1}}=\left[\begin{array}{ccc}{0} & {1} & {0} \\ {1} & {1} & {1} \\ {1} & {0} & {0}\end{array}\right] \text { and } \mathbf{M}_{R_{2}}=\left[\begin{array}{ccc}{0} & {1} & {0} \\ {0} & {1} & {1} \\\ {1} & {1} & {1}\end{array}\right]$$ Find the matrices that represent $$\begin{array}{llll}{\text { a) } R_{1} \cup R_{2}} & {\text { b) } R_{1} \cap R_{2}} & {\text { c) } R_{2} \circ R_{1}} \\ {\text { d) } R_{1} \circ R_{1}} & {\text { e) } R_{1} \oplus R_{2}}\end{array}$$

Consider the equivalence relation from Example 2 namely, \(R=\\{(x, y) | x-y \text { is an integer }\\} .\) a) What is the equivalence class of 1 for this equivalence relation? b) What is the equivalence class of 1\(/ 2\) for this equivalence relation?

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.

Show that the relation of logical equivalence on the set of all compound propositions is an equivalence relation. What are the equivalence classes of \(\mathbf{F}\) and of \(\mathbf{T} ?\)
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