Question

Describe the partition of \(\mathbb{Z}\) resulting from the equivalence relation \(\equiv(\bmod 4)\)

Short answer

The partition of \(\mathbb{Z}\) by \(\equiv(\bmod 4)\) will form four subsets. [0] = \(\ldots,-8, -4, 0, 4, 8, \ldots\), [1] = \(\ldots,-7, -3, 1, 5, 9, \ldots\), [2] = \(\ldots,-6, -2, 2, 6, 10, \ldots\), and [3] = \(\ldots,-5, -1, 3, 7, 11, \ldots\). Each integer in \(\mathbb{Z}\) belongs to exactly one of these subsets

Step-by-step solution

Understanding equivalence relation

The equivalence relation of the modulo operator groups integers together based on the remainder they give when divided by a certain number. In this problem, the modulo operator is four; hence integers will be grouped together based on their remainder when divided by 4. So, there are 4 classes - [0], [1], [2], and [3] as a result of the operation.

Describe the partition of Integers using \(\equiv(\bmod 4)\)

Illustrating the sets of integers for the aforementioned classes: The set [0] will contain all the integers that leave a remainder 0 when divided by 4, i.e., all multiples of 4: \(\ldots,-8, -4, 0, 4, 8, \ldots\). The set [1] will contain all integers that leave a remainder of 1 when divided by 4, i.e., integers that are 1 more than a multiple of 4: \(\ldots,-7, -3, 1, 5, 9, \ldots\). The set [2] has integers that are 2 more than a multiple of 4: \(\ldots,-6, -2, 2, 6, 10, \ldots\). The set [3] includes integers that are 3 more than a multiple of 4: \(\ldots,-5, -1, 3, 7, 11, \ldots\). The distinct sets together make up the whole set of integers, without any overlaps or empty sets.

Key concepts

Modular Arithmetic

Modular arithmetic is often referred to as 'clock arithmetic' because of the way it cycles numbers similar to the hours on a clock. In essence, modular arithmetic deals with the remainder when an integer is divided by a positive integer known as the modulus. For example, if we say '7 modulo 4', denoted as \(7 \bmod 4\), we are interested in what remains after dividing 7 by 4, which is 3.

This concept is fundamental in various fields such as cryptography, computer science, and number theory. It simplifies arithmetic for large numbers by dealing with their remainders, thus reducing the complexity of calculations.

In educational contexts, mastering modular arithmetic is essential for understanding more complex concepts like encryption algorithms or the behavior of numerical systems with fixed bounds.

Integer Partition

An integer partition, in terms of modular arithmetic, can be envisioned as sorting integers into distinct groups where each group, or partition, is defined by a particular property. For the exercise at hand, where the equivalence relation uses \(\equiv(\bmod 4)\), integers are partitioned into four distinct sets based on their remainders when divided by 4.

It is quite similar to putting balls into boxes where each box has a label from 0 to 3 (when using modulo 4). Every integer is like a ball that, when divided by 4, falls into one of these labeled boxes based on the remainder. The concept of integer partitioning is a springboard into topics such as combinatorics, number theory, and even helps in breaking problems into smaller, more manageable sub-problems.

Congruence Modulo

Congruence modulo is a way of expressing that two numbers are equivalent under modular arithmetic. For instance, if we take the number 9 and 5 modulo 4, they both leave a remainder of 1 upon division by 4, hence they are congruent modulo 4, and we write \(9 \equiv 5 (\bmod 4)\).

This concept is closely related to the idea of equivalence classes or partitions mentioned in the textbook exercise: each set [0], [1], [2], and [3], represents an equivalence class where all the members are congruent to each other modulo 4.

Digging deeper into congruence modulo not only aids in solving modular arithmetic problems but also lays a strong foundation for understanding algebraic structures such as rings and fields, which have vast applications in advanced mathematics and related disciplines.