Question

Identify the equivalence classes of \(\mathrm{N}\) for the following relations: (a) \(\equiv(\bmod 4)\) (b) \(\equiv(\bmod 6)\)

Short answer

Modulo 4: {[0], [1], [2], [3]}; Modulo 6: {[0], [1], [2], [3], [4], [5]}.

Step-by-step solution

Understand the Problem

We are tasked with finding the equivalence classes for the natural numbers under the given modulo relations. An equivalence class groups numbers that are equivalent to each other under a given relation. Here, we need to consider modulo 4 and modulo 6.

Define Modulo Equivalence

The equivalence class of a number under modulo relation contains all numbers that leave the same remainder when divided by the given modulus. For a number \(a\) and modulus \(n\), the equivalence class is \([a]_n = \{ x \in \mathbb{N} \,|\, x \equiv a \pmod{n} \}\).

Find Equivalence Classes for \( \equiv (\mod 4) \)

For \(\equiv(\mod 4)\), consider the remainders when a number is divided by 4: 0, 1, 2, and 3. Thus, the equivalence classes are:- \([0]_4 = \{0, 4, 8, 12, \ldots\}\) - \([1]_4 = \{1, 5, 9, 13, \ldots\}\)- \([2]_4 = \{2, 6, 10, 14, \ldots\}\)- \([3]_4 = \{3, 7, 11, 15, \ldots\}\).

Find Equivalence Classes for \( \equiv (\mod 6) \)

For \(\equiv(\mod 6)\), the possible remainders are 0, 1, 2, 3, 4, and 5. Therefore, the equivalence classes are:- \([0]_6 = \{0, 6, 12, 18, \ldots\}\)- \([1]_6 = \{1, 7, 13, 19, \ldots\}\)- \([2]_6 = \{2, 8, 14, 20, \ldots\}\)- \([3]_6 = \{3, 9, 15, 21, \ldots\}\)- \([4]_6 = \{4, 10, 16, 22, \ldots\}\)- \([5]_6 = \{5, 11, 17, 23, \ldots\}\).

Verify Equivalence Classes

Verify that each equivalence class contains numbers that are congruent to each other under the modulo relation. Each number in an equivalence class differs from another by a multiple of the modulus.

Key concepts

Modulo Operation

The modulo operation is like finding the remainder when one number is divided by another. It's often written as "mod", and you might see it like this: \( a \mod n \). Here \( a \) is the number you're working with and \( n \) is the divisor. The result of \( a \mod n \) is the remainder, which is what's left over after dividing \( a \) by \( n \). This operation has some special properties that are useful in math, such as grouping numbers into equivalence classes.

• If \( a = 17 \) and \( n = 5 \), then \( 17 \mod 5 \) gives a remainder of 2, because 5 goes into 17 three times (15) with 2 left over.
• Modulo operation is cyclical because it repeats patterns every \( n \) numbers, like a clock that repeats every 12 hours.
• It helps in constructing equivalence classes by grouping numbers that share the same remainder.

These basic ideas are crucial in understanding how the modulo operation helps form equivalence classes.

Natural Numbers

Natural numbers are the set of positive integers starting from 0 and increasing without end: 0, 1, 2, 3, and so on. They do not include negative numbers or fractions. Natural numbers are often symbolized by \( \mathbb{N} \), and they are foundational in mathematics for counting and ordering.

• Natural numbers begin from 0 in modern usage, but sometimes they start from 1 depending on the context.
• They are endless, meaning you can always add 1 to any natural number and still have a natural number.
• Natural numbers are closed under addition and multiplication, which means that adding or multiplying any two natural numbers results in another natural number.

When we discuss equivalence classes in natural numbers, we are primarily focused on how they are partitioned under modulo operations.

Congruence Relation

Congruence relations stem from the modulo operation and are a way of grouping numbers together based on their remainders. When two numbers have the same remainder when divided by a number, they are said to be congruent modulo that number. We write \( a \equiv b \pmod{n} \) to mean that \( a \) and \( b \) have the same remainder when divided by \( n \).

• This concept can be visualized by imagining different numbers on a number line loop, like a circular track marked at remainder points.
• For example, \( 17 \equiv 2 \pmod{5} \) because when you calculate \( 17 \mod 5 \) and \( 2 \mod 5 \), both give the remainder 2.
• The key function of congruence relations is to create equivalence classes, which are sets of numbers that share this property of congruence.

Understanding congruence relations proves useful in many areas of mathematics, including number theory and algebra, as they simplify complex problems by reducing them to patterns of remainders.