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Chapter 2: Problem 11

For \(a, b \in \mathrm{Z}\), define \(a \sim b\) if and only if \(a^{2}-b^{2}\) is divisible by 3 . (a) \([\mathrm{BB}]\) Prove that \(\sim\) defines an equivalence relation on \(\mathrm{Z}\). (b) What is \(\overline{0}\) ? What is \(\overline{1}\) ? (c) What is the partition of \(Z\) determined by this equivalence relation?

Short Answer

Expert verified
The relation is an equivalence relation. \( \overline{0} = \{ b \equiv 0 \pmod{3} \} \), \( \overline{1} = \{ b \equiv 1 \pmod{3} \} \). The partition is \( \{ \overline{0}, \overline{1}, \overline{2} \} \).

Step by step solution

01

Understand the Problem

We are given a relation \( a \sim b \) defined such that \( a^2 - b^2 \) is divisible by 3. We need to prove this is an equivalence relation, and determine the equivalence classes \( \overline{0} \) and \( \overline{1} \), as well as the partition of \( \mathbb{Z} \).
02

Prove Reflexivity

Reflexivity means for any integer \( a \), we have \( a \sim a \). Since \( a^2 - a^2 = 0 \), which is divisible by 3, \( a \sim a \) holds true.
03

Prove Symmetry

Symmetry means if \( a \sim b \), then \( b \sim a \). Assuming \( a^2 - b^2 \) is divisible by 3, the expression \( b^2 - a^2 = -(a^2 - b^2) \) is also divisible by 3. Thus \( b \sim a \).
04

Prove Transitivity

Transitivity means if \( a \sim b \) and \( b \sim c \), then \( a \sim c \). Assume \( a^2 - b^2 \equiv 0 \pmod{3} \) and \( b^2 - c^2 \equiv 0 \pmod{3} \). Then \( a^2 - c^2 = (a^2 - b^2) + (b^2 - c^2) \equiv 0 \pmod{3} \). Hence \( a \sim c \).
05

Determine \(\overline{0}\)

\( \overline{0} \) is the set of integers \( b \) such that \( 0 \sim b \) or \( 0^2 - b^2 \) is divisible by 3. This means \( b^2 \equiv 0 \pmod{3} \), i.e., \( b \equiv 0 \pmod{3} \). \( \overline{0} = \{ b \in \mathbb{Z} : b \equiv 0 \pmod{3} \} \).
06

Determine \(\overline{1}\)

\( \overline{1} \) is the set of integers \( b \) such that \( 1 \sim b \) or \( 1^2 - b^2 \) is divisible by 3. This means \( b^2 \equiv 1 \pmod{3} \), i.e., \( b \equiv \pm 1 \pmod{3} \). \( \overline{1} = \{ b \in \mathbb{Z} : b \equiv 1 \pmod{3} \} \cup \{ b \in \mathbb{Z} : b \equiv 2 \pmod{3} \} \).
07

Determine the Partition

The partition determined by this equivalence relation consists of the equivalence classes \( \overline{0} = \{ b : b \equiv 0 \pmod{3} \} \), \( \overline{1} = \{ b : b \equiv 1 \pmod{3} \} \), and \( \overline{2} = \{ b : b \equiv 2 \pmod{3} \} \).

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Key Concepts

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

Reflexivity
Reflexivity is one of the foundational properties we need to check when determining if a relation is an equivalence relation. For a relation to be reflexive, every element must be related to itself. Let's consider an integer \(a\).
If we say that \(a\sim a\), it means that \(a^2 - a^2\) should be divisible by 3.\(a^2 - a^2 = 0\), and zero is always divisible by 3.
This means every integer is related to itself under the relation defined by divisibility.Therefore, the property of reflexivity is satisfied. Here, understanding reflexivity is intuitive because it checks if a is "equally" related to itself, holding true for all elements of \(\mathbb{Z}\).
This property ensures that the relation doesn't leave out the simplest self-comparison scenario, where the element is its own counterpart.
Symmetry
Symmetry is another crucial characteristic in defining an equivalence relation.
For a relation to be symmetric, whenever \(a\sim b\), it should also imply that \(b\sim a\). Consider the given relation where \(a \sim b\) if \(a^2 - b^2\) is divisible by 3.
If this holds true, then it must equally hold for \(b^2 - a^2\), which can be simplified to \(-(a^2 - b^2)\), still effectively divisible by 3.Thus, if \(a\) is related to \(b\), we've shown that \(b\) is equally related back to \(a\), demonstrating symmetry. This property ensures reversibility in the relationship between any two elements.
Symmetry allows the relation to be bidirectional, affirming that the order of the elements doesn't affect their connection in terms of the relation.
Transitivity
Transitivity is the third property essential for confirming whether a relation is an equivalence.
For a relation to be transitive, the sequence \(a\sim b\) and \(b\sim c\) must guarantee \(a\sim c\). This can be understood better by example.
Given \(a\sim b\) means \(a^2 - b^2\) is divisible by 3, and \(b\sim c\) means \(b^2 - c^2\) is also divisible by 3.Adding these up gives \((a^2 - b^2) + (b^2 - c^2) = a^2 - c^2\).
This net difference is similarly divisible by 3, thus confirming \(a\sim c\). This property logically extends the connection through an intermediate element.
Transitivity ensures that the relation maintains a consistent connection all the way through multiple related elements, allowing seamless transition through a chain of relations.

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Most popular questions from this chapter

(a) Let \(A=Z^{2}\) and, for \(a=\left(a_{1}, a_{2}\right)\) and \(b=\left(b_{1}, b_{2}\right)\) in \(A\), define \(a \preceq b\) if and only if \(a_{1} \leq b_{1}\) and \(a_{1}+a_{2} \leq b_{1}+b_{2} .\) Prove that \(\leq\) is a partial order on \(A\). Is this partial order a total order? Justify your answer with a proof or a counterexample. (b) Generalize the result of part (a) by defining a partial order on the set \(Z^{\prime \prime}\) of \(n\) -tuples of integers. (No proof is required.)

The sets \(\\{1\\},\\{2\\},\\{3\\},\\{4\\},\\{5\\}\) are the equivalence classes for a well-known equivalence relation on the set \(S=\\{1,2,3,4,5\\}\). What is the usual name for this equivalence relation?

\([\mathrm{BB}]\) For \(a, b \in \mathrm{R}\), define \(a \sim b\) if and only if \(a-b \in \mathrm{Z}\). (a) Prove that \(\sim\) defines an equivalence relation on \(\mathrm{Z}\). (b) What is the equivalence class of \(5 ?\) What is the equivalence class of \(5 \frac{1}{2}\) ? (c) What is the quotient set determined by this equivalence relation?

(a) [BB] Prove that a glb of two elements in a poset \((A, \preceq)\) is unique whenever it exists. (b) Prove that a lub of two elements in a poset \((A, \preceq)\) is unique whenever it exists.

Let \(A\) be the set of books for sale in a certain university bookstore and assume that among these are books with the following properties. (a) [BB] Suppose \((a, b) \in \mathcal{R}\) if and only if the price of book \(a\) is greater than or equal to the price of book \(b\) and the length of \(a\) is greater than or equal to the length of \(b .\) Is \(\mathcal{R}\) reflexive? Symmetric? Antisymmetric? Transitive? (b) Suppose \((a, b) \in \mathcal{R}\) if and only if the price of \(a\) is greater than or equal to the price of \(b\) or the length of \(a\) is greater than or equal to the length of \(b .\) Is \(\mathcal{R}\) reflexive? Symmetric? Antisymmetric? Transitive?
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