Question

Find the elements in the relation "have the same remainder when divided by \(8^{\prime \prime}\) if the relation is defined on \(\\{1,2,3, \ldots, 24,25\\} .\) Also, find the distinct equivalence classes of this equivalence relation.

Short answer

There are 8 equivalence classes based on remainders from 0 to 7.

Step-by-step solution

Understanding the Problem

We need to find elements from the set \(\{1,2,3, \ldots, 25\}\) that have the same remainder when divided by 8. This establishes an equivalence relation where two numbers are equivalent if they leave the same remainder when divided by 8.

Determine Remainders

Divide each element from 1 to 25 by 8 and find the remainder for each. For example, 1 divided by 8 gives a remainder of 1, 2 divided by 8 gives a remainder of 2, and so on.

Group Elements by Remainder

Group the elements according to their remainders. Elements with the same remainder form an equivalence class.

List of Equivalence Classes

Determine and list out the distinct equivalence classes:- Remainder 0: \(\{8, 16, 24\}\)- Remainder 1: \(\{1, 9, 17, 25\}\)- Remainder 2: \(\{2, 10, 18\}\)- Remainder 3: \(\{3, 11, 19\}\)- Remainder 4: \(\{4, 12, 20\}\)- Remainder 5: \(\{5, 13, 21\}\)- Remainder 6: \(\{6, 14, 22\}\)- Remainder 7: \(\{7, 15, 23\}\)

Verify Completeness

Check that all numbers from 1 to 25 are included and covered exactly once in one of the equivalence classes. This ensures all numbers are accounted for correctly.

Key concepts

Equivalence Classes

Equivalence classes are a fundamental concept in mathematics, particularly in the study of equivalence relations. In essence, an equivalence class groups elements that are considered equivalent under a given relation. In the context of our problem, elements are grouped by their remainders when divided by 8. Each group, or class, contains numbers that exhibit this kind of equivalence.
For example, when we divide the set \{1, 2, 3, \ldots, 25\} by 8, we form equivalence classes like \(\{1, 9, 17, 25\}\) for remainder 1, and so on. These classes help us see which numbers share the same result in this division operation.

Equivalence classes must fulfill three properties:

• Reflexivity: Every element is equal to itself, and thus in the same equivalence class.
• Symmetry: If one element relates to a second, the second relates to the first, making them in the same class.
• Transitivity: If an element relates to a second, and the second to a third, the first relates to the third, placing all in the same class.

Understanding these properties ensures clarity and confidence in identifying equivalence classes.

Remainders

Remainders play a crucial role in dividing numbers, especially when dealing with problems involving equivalence relations like ours. The remainder is what is left over after a number is divided by another. In simpler terms, for any integer division, \[ dividend = divisor \times quotient + remainder \]
When we talk about dividing a number by 8, as our exercise prompts, the remainder can be any number from 0 to 7. For example, the division of 25 by 8 gives us a quotient of 3 with a remainder of 1. Thus, 25 belongs to the equivalence class of numbers that also have a remainder of 1 when divided by 8.

Working with remainders helps us:

• Identify relationships between numbers based on how they "fit" within the divisor.
• Group numbers under common equivalences, such as in equivalence classes.

Recognizing and calculating remainders is essential for solving many problems in number theory and algebra.

Modular Arithmetic

Modular arithmetic is often referred to as arithmetic "modulo" some number, and it is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, known as the modulus. In our exercise, the modulus is 8.
For instance, saying "25 is congruent to 1 modulo 8" is a way of saying that when 25 is divided by 8, the remainder is 1.
In mathematical terms:\[ a \equiv b \pmod{m} \]means that \(a - b\) is a multiple of the modulus \(m\). In our case, \(25 \equiv 1 \pmod{8}\).

Modular arithmetic is key in various applications:

• Coding theory and cryptography rely heavily on mods.
• It simplifies calculations by focusing only on remainders.
• It helps define equivalence relations, as seen in our exercise.

This arithmetic is widespread because it neatly and efficiently generalizes various principles of arithmetic.

Set Theory

Set theory is a branch of mathematical logic that explores collections of objects, known as sets. A set is typically defined as a collection of distinct elements. In the context of this exercise, the set of numbers \{1, 2, 3, \ldots, 25\} is under examination.
Using set theory to define equivalence classes allows us to understand the partitioning of the set into smaller, distinct groups based on shared properties, like remainders when divided by 8.

Key concepts of set theory relevant to this problem include:

• Elements: The individual numbers within our set \(\{1, 2, 3, \ldots, 25\}\).
• Subset: Each equivalence class is a subset of the original set, containing elements sharing the same remainder.
• Partition: The complete division of the original set into equivalence classes with each element in exactly one class.

Understanding foundations of set theory provides a clear structure in organizing and manipulating data, essential for both discrete mathematics and computer science.