Question

Congruence modulo 5 is a relation on the set \(A=\mathbb{Z} .\) In this relation \(x R y\) means \(x \equiv y(\bmod 5) .\) Write out the set \(R\) in set-builder notation.

Short answer

The set R in set-builder notation is \(R = \{(x, y) \in \mathbb{Z} x \mathbb{Z} | x - y\ is\ divisible\ by\ 5\}\)

Step-by-step solution

Understanding Congruence modulo

Our first step is to understand what 'x R y' means - x is congruent to y modulo 5. Modulo operation finds the remainder after division of one number by another (also called modulus). In this case, the numbers x and y leave the same remainder when divided by 5.

Write out the relation

We now write out this relationship using set-builder notation.

The final step

The relation R on A can be written as \(R = \{(x, y) \in \mathbb{Z} x \mathbb{Z} | x - y\ is\ divisible\ by\ 5\}\). This states that for any pair (x, y) that belongs to the set of integers, x is related to y if and only if the difference between x and y is divisible by 5, which is congruence modulo 5.

Key concepts

Set-Builder Notation

Set-builder notation is a shorthand used in mathematics to describe a set by stating the properties that its members must satisfy.
For example, to express the set of all integers divisible by 5, we could write it in set-builder notation as \( \{ x \in \mathbb{Z} | x \equiv 0(\bmod 5) \} \).
This notation is compact and precise, letting us capture complex ideas succinctly. When the textbook problem asks to use set-builder notation to describe the relationship where \( x \equiv y(\bmod 5) \), the student is being guided to think about the defining characteristic of that relationship – namely, that the difference between any two numbers in the set is divisible by 5.
Therefore, the relation \( R \) can be concisely expressed using set-builder notation to show all possible integer pairs that satisfy the given condition.

Modular Arithmetic

Modular arithmetic is a system of arithmetic for integers, where numbers 'wrap around' after they reach a certain value—the modulus. It's like a mathematical version of a clock: after 12:59, it doesn’t turn to 13:00 but to 1:00.
In this system, we say two numbers are congruent modulo a given modulus if they have the same remainder when divided by that modulus. For instance, with modulus 5, 11 is congruent to 1 because both leave a remainder of 1 when divided by 5 (11 % 5 = 1, 1 % 5 = 1).
Using the language of modular arithmetic, the relation described in the exercise could be understood as \( x \equiv y (\bmod 5) \), signifying that \( x \) and \( y \) are part of an equivalence class defined by their shared remainder when divided by five.

Number Theory

Fundamentals of Number Theory

Number theory is a branch of mathematics that deals with the properties and relationships of numbers, primarily the integers. It has applications in cryptography, computer science, and problem-solving.
Within number theory, the concept of congruence, as demonstrated in the exercise with congruence modulo 5, is a cornerstone. It explores the idea that integer relationships can be characterized by their residues when divided by a modulus.
Understanding the underpinning principles of number theory can help students approach problems involving prime numbers, divisibility, and modular arithmetic with greater ease and insight.
By integrating number theory concepts within the context of congruence and modular arithmetic, the exercise shows one of the ways this field of mathematics can be practically applied.

Integer Sets

Understanding Integer Sets

Integer sets are simply sets which consist of integers, such as the set of all whole numbers, positive and negative (including zero). These sets are foundational in number theory and are often denoted by the symbol \( \mathbb{Z} \).
When discussing relationships like congruence modulo within integer sets, we're examining how integers can be categorized or grouped according to specific properties, such as their remainders when divided by a particular number.
The set \( \mathbb{Z} \) is vital in formulating problems in number theory because it represents the universal set of integers in which relations and operations are generally considered. In the textbook problem, the relation \( R \) comprises ordered pairs of integers that demonstrate the specified congruence property.
Comprehending integer sets is critical for students to understand modular relationships and other concepts in number theory.