Question

Prove that if \(s \in[m]_{n}\) and \(t \in[k]_{n}\) then \(s \times t \in[m\) \(\times k] n\).

Short answer

The product of two elements from equivalence classes modulo-n will also be an element of an equivalence class modulo-n. Specifically, if \(s \in[m]_{n}\) and \(t \in[k]_{n}\), then their product \(s \times t\) will belong to the equivalence class modulo-n represented by the product of the representatives of the original two classes, \(m \times k\).

Step-by-step solution

Introduction of arbitrary elements

Let's introduce two arbitrary elements from the equivalence classes \(s\) and \(t\). These can be represented as \(s = m + nx\) and \(t = k + ny\) where \(x\) and \(y\) are integers.

Multiply the elements

Now we multiply these elements together. \(s \times t = (m + nx) \times (k + ny) = mk + n(mx + ky + nxy)\). We can see the product can be written in the form \(a + nb\) where \(a = mk\) and \(b = mx + ky + nxy\), which are both integers since \(m\), \(k\), \(x\), and \(y\) are integers and \(n\) is a natural number.

Conclude the proof

Since the product \(s \times t = mk + n(mx + ky + nxy)\) is of the form \(a + nb\) where \(a\) and \(b\) are integers, this implies that \(s \times t\) is an element of the equivalence class modulo n, specifically the \(mk modulo-n\) class. Thus, it is proven that if \(s \in[m]_{n}\) and \(t \in[k]_{n}\) then \(s \times t \in[m \times k]_{n}\).

Key concepts

Equivalence Classes

In number theory, equivalence classes play a crucial role in simplifying complex calculations. When dealing with equivalence classes, we're essentially grouping integers that share a specific relationship, often represented by congruences. For instance, when we say two numbers, say 12 and 5, are congruent modulo 7, this means that both 12 and 5 leave the same remainder when divided by 7, which is 5 in this case.
This congruence can be denoted as: \(12 \equiv 5 \pmod{7}\). Likewise, equivalence classes are written in the form \([m]_n\), grouping all numbers that are equivalent to \(m\) under modulo \(n\).

• An equivalence class modulo \(n\) is a set of numbers that have the same remainder when divided by \(n\).
• Properties of equivalence classes allow us to perform operations like addition and multiplication under modulo constraints more easily.

Equivalence classes are fundamental when working with modular arithmetic as they serve the basis for understanding how numbers interact in the modular system.

Modular Arithmetic

Modular arithmetic is often likened to the arithmetic of a clock. It concerns integer division and focuses on remainders rather than the quotient. This arithmetic system is essential in various fields like cryptography, computer science, and even in solving certain equations in physics and engineering. For example, in a 12-hour clock, after 11, the next number is not 12 but 0. This demonstrates the concept of modulus in modular arithmetic.
A common example of modular arithmetic is congruence relation, expressed as \(a \equiv b \pmod{n}\), which means \(a\) and \(b\) have the same remainder when divided by \(n\).

• Operations with modular arithmetic follow specific rules, such as for addition, \((a + b) \mod n = (a \mod n + b \mod n) \mod n\).
• Multiplication follows the rule: \((a \times b) \mod n = ((a \mod n) \times (b \mod n)) \mod n\).

Modular arithmetic simplifies calculations that would otherwise be cumbersome if dealt only in integers, making it invaluable in testing divisibility and properties of numbers through equivalence classes.

Proof Techniques

Proof techniques are essential tools in mathematics, allowing us to rigorously demonstrate the truth of statements. When working with equivalence classes and modular arithmetic, we often engage with proofs that involve leveraging properties of integers, congruences, or specific algebraic manipulations.
For instance, to prove statements involving equivalence classes, a common technique is introducing elements represented in terms of integers. This technique was demonstrated in the problem step: expressing arbitrary representatives \(s = m + nx\) and \(t = k + ny\) helps in understanding how multiplication within classes works.

• One common proof technique is direct proof, where the argument is constructed logically from known principles or axioms.
• Another approach is proof by contradiction, assuming the opposite of what you want to prove and showing that it leads to an inconsistency.

The exercise solution exemplified a direct proof, showcasing multiplication within equivalence classes and broken down into simpler integer operations. Applying these proof techniques ensures conclusions are reliable and widely accepted in mathematical discourse.