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Chapter 7: Problem 4

Determine all of the congruence classes for the relation of congruence modulo 5 on the set of integers.

Short Answer

Expert verified
The congruence classes for the relation of congruence modulo 5 on the set of integers are: 1. \([0]\): \(\{0, \pm5, \pm10, \pm15, ...\}\) 2. \([1]\): \(\{\pm1, \pm6, \pm11, \pm16, ...\}\) 3. \([2]\): \(\{\pm2, \pm7, \pm12, \pm17, ...\}\) 4. \([3]\): \(\{\pm3, \pm8, \pm13, \pm18, ...\}\) 5. \([4]\): \(\{\pm4, \pm9, \pm14, \pm19, ...\}\)

Step by step solution

01

Understanding modulo and congruence classes

A congruence class modulo 5 is a set of integers that have the same remainder when divided by 5. The congruence relation is denoted as "鈮�", for example: a 鈮� b (mod 5) means that a and b have the same remainder when divided by 5.
02

Analyzing the remainders

Since we are working with modulo 5, there can be only 5 possible remainders: 0, 1, 2, 3, and 4. So, there will be 5 distinct congruence classes: [0], [1], [2], [3], and [4].
03

Forming the congruence classes

Now, we will form the congruence classes by finding the integers that give the same remainder when divided by 5. 1. For [0]: The integers that have a remainder of 0 when divided by 5 are: ..., -10, -5, 0, 5, 10, 15,... 2. For [1]: The integers that have a remainder of 1 when divided by 5 are: ..., -9, -4, 1, 6, 11, 16,... 3. For [2]: The integers that have a remainder of 2 when divided by 5 are: ..., -8, -3, 2, 7, 12, 17,... 4. For [3]: The integers that have a remainder of 3 when divided by 5 are: ..., -7, -2, 3, 8, 13, 18,... 5. For [4]: The integers that have a remainder of 4 when divided by 5 are: ..., -6, -1, 4, 9, 14, 19,...
04

Solution

The congruence classes for the relation of congruence modulo 5 on the set of integers are: 1. [0]: {0, 卤5, 卤10, 卤15, ...} 2. [1]: {卤1, 卤6, 卤11, 卤16, ...} 3. [2]: {卤2, 卤7, 卤12, 卤17, ...} 4. [3]: {卤3, 卤8, 卤13, 卤18, ...} 5. [4]: {卤4, 卤9, 卤14, 卤19, ...}

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Key Concepts

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

Modular Arithmetic
Modular arithmetic is a system of arithmetic for integers, where numbers wrap around after they reach a certain value鈥攖he modulus. Imagine a clock with 12 hours; when you go past 12, you start again from 1. That's modular arithmetic with a modulus of 12. In the context of the exercise, the modulus is 5, so we consider numbers as being 'cyclic' after every 5 units.

When we perform calculations such as addition, subtraction, and multiplication using modular arithmetic, we're essentially interested in the remainder after dividing by the modulus. If we say '7 plus 5, modulo 5', we want to know what remains after we've 'wrapped around' the modulus. Here, 7 + 5 is 12, and when we divide 12 by 5, we're left with a remainder of 2, so 7 plus 5 is congruent to 2 mod 5.
Integer Remainder Division
Integer remainder division is an operation that divides an integer by another integer, yielding a quotient and a remainder. This is the division most of us learned in our early school years. The remainder is what is left over after dividing the first number by the second as many times as possible without going into negative numbers.

In our example of dividing numbers by 5, the remainders form a finite set of possibilities: 0, 1, 2, 3, and 4. These numbers represent the possible outcomes of the modulo operation when wereferred to in the congruence classes. For instance, 11 divided by 5 gives a quotient of 2 and a remainder of 1. In this case, we could say 11 mod 5 equals 1.
Congruence Relation
A congruence relation is an equivalence relation that separates numbers into distinct classes based on their remainders when divided by a given modulus. Two integers are said to be congruent modulo n if their difference is a multiple of n, which means they have the same remainder upon division by n.

Using a modulus of 5, we can say that 11 and 6 are congruent, written as 11 鈮� 6 (mod 5), because both leave a remainder of 1 when divided by 5. The congruence classes represented in the exercise solution are sets of all numbers that are congruent to each other modulo 5. Each class can be thought of as a 'family' of numbers that share a common characteristic: the remainder they leave when divided by 5.

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Most popular questions from this chapter

The set \(\mathbb{Z}_{n}\) is a finite set, and hence one way to prove things about \(\mathbb{Z}_{n}\) is to simply use the \(n\) elements in \(\mathbb{Z}_{n}\) as the \(n\) cases for a proof using cases. For example, if \(n \in \mathbb{Z},\) then in \(\mathbb{Z}_{4},[n]=[0]\), \([n]=[1],[n]=[2],\) or \([n]=[3]\) (a) Prove that if \(n \in \mathbb{Z},\) then in \(\mathbb{Z}_{4},[n]^{2}=[0]\) or \([n]^{2}=[1] .\) Use this to conclude that in \(\mathbb{Z}_{4},\left[n^{2}\right]=[0]\) or \(\left[n^{2}\right]=[1]\) (b) Translate the equations \(\left[n^{2}\right]=[0]\) and \(\left[n^{2}\right]=[1]\) in \(\mathbb{Z}_{4}\) into congruences modulo 4 . (c) Use a result in Exercise (12) to determine the value of \(r\) so that \(r \in \mathbb{Z}\) \(0 \leq r<3,\) and $$ 104257833259 \equiv r(\bmod 4) $$ That is, \([104257833259]=[r]\) in \(\mathbb{Z}_{4}\). (d) Is the natural number 104257833259 a perfect square? Justify your conclusion.

Let \(U\) be a nonempty set, and let \(R\) be the "element of" relation from \(U\) to \(\mathcal{P}(U) .\) That is, $$ R=\\{(x, S) \in U \times \mathcal{P}(U) \mid x \in S\\} $$ (a) What is the domain of this "element of' relation, \(R ?\) (b) What is the range of this "element of" relation, \(R ?\) (c) Is \(R\) a function from \(U\) to \(\mathcal{P}(U)\) ? Explain.

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be defined by \(f(x)=x^{2}-4\) for each \(x \in \mathbb{R}\). Define a relation \(\sim\) on \(\mathbb{R}\) as follows: For \(a, b \in \mathbb{R}, a \sim b\) if and only if \(f(a)=f(b)\). (a) Is the relation \(\sim\) an equivalence relation on \(\mathbb{R} ?\) Justify your conclusion. (b) Determine all real numbers in the set \(C=\\{x \in \mathbb{R} \mid x \sim 5\\}\).

Let \(A=\mathbb{Z} \times(\mathbb{Z}-\\{0\\}) .\) That is, \(A=\\{(a, b) \in \mathbb{Z} \times \mathbb{Z} \mid b \neq 0\\} .\) Define the relation \(\approx\) on \(A\) as follows: For \((a, b),(c, d) \in A,(a, b) \approx(c, d)\) if and only if \(a d=b c\) (a) Prove that \(\approx\) is an equivalence relation on \(A\). (b) Why was it necessary to include the restriction that \(b \neq 0\) in the definition of the set \(A ?\) - (c) Determine an equation that gives a relation between \(a\) and \(b\) if \((a, b) \in\) \(A\) and \((a, b) \approx(2,3)\) (d) Determine at least four different elements in [(2,3)] , the equivalence class of (2,3) . (e) Use set builder notation to describe \([(2,3)],\) the equivalence class of (2,3)

Let \(A=\\{a, b, c\\}\). For each of the following, draw a directed graph that represents a relation with the specified properties. (a) A relation on \(A\) that is symmetric but not transitive (b) A relation on \(A\) that is transitive but not symmetric (c) A relation on \(A\) that is symmetric and transitive but not reflexive on \(A\) (d) A relation on \(A\) that is not reflexive on \(A\), is not symmetric, and is not transitive (e) A relation on \(A\), other than the identity relation, that is an equivalence relation on \(A\)
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