Problem 1 - A Book of Abstract Algebra

Question

For each integer \(r \in\\{0,1,2,3,4\\}\), let \(A_{r}\) be the set of all the integers which leave a remainder of \(r\) when divided by 5 . (That is, \(x \in A_{r}\) iff \(x=5 q+r\) for some integer \(q\).) Prove: \(\left\\{A_{0}, A_{1}, A_{2}, A_{3}, A_{4}\right\\}\) is a partition of \(\mathbb{Z}\)

Step-by-Step Solution

Answer

Sets \(\{A_0, A_1, A_2, A_3, A_4\}\) partition \(\mathbb{Z}\) by non-overlapping and collectively covering all integers.

Step 1: Define a Partition

To prove that sets \(\{A_0, A_1, A_2, A_3, A_4\}\) form a partition of \(\mathbb{Z}\), we need to show that every integer belongs to exactly one of these sets. A partition of a set is a division of elements into non-overlapping subsets such that every element belongs to one subset.

Step 2: Define Set Membership

Each set \(A_r\) contains integers that give a specific remainder \(r\) when divided by 5. This is determined by the equation \(x = 5q + r\), where \(r\) can be 0, 1, 2, 3, or 4. Hence, every integer \(x\) must belong to one of these sets according to their remainder upon division by 5.

Step 3: Non-overlapping Segments

Next, we must show that these sets do not overlap, meaning no integer can belong to more than one set. If two sets \(A_r\) and \(A_s\) overlap, there would be a number that leaves different remainders when divided by 5, which is impossible.

Step 4: Entirely Covers \(\mathbb{Z}\)

Finally, the union of all these sets should cover the entire set of integers. Since every integer has a remainder of either 0, 1, 2, 3, or 4 when divided by 5, the sets \(\{A_0, A_1, A_2, A_3, A_4\}\) together cover all integers without omission.

Key Concepts

Remainder Division

Remainder division is a technique used to express the division of two integers. When an integer \(x\) is divided by another integer \(n\), it results in a quotient \(q\) and a remainder \(r\). This relationship can be mathematically expressed as:
\[ x = nq + r \]where \(0 \leq r < n\).
The remainder \(r\) is what remains after the division process. It indicates how much of \(x\) is left unaccounted for after whole multiples of \(n\) have been taken out.In the context of our problem, the integer \(x\) divided by 5 results in remainders ranging from 0 to 4. This means:

• If \(r = 0\), \(x\) is perfectly divisible by 5.
• If \(r = 1\), there is 1 left after division.
• If \(r = 2\), there are 2 left after division, and so on.

Understanding remainder division helps better grasp integer partition scenarios, where integers are grouped based on their remainders when divided by a particular number.

Integer Sets

Integer sets are collections of integers that share common properties or characteristics. In mathematics, these sets often help organize numbers for easier manipulation and examination. For the exercise at hand, we're dealing with specific integer sets defined by their remainders after division by 5.
Set \(A_r\), for example, consists of all integers \(x\) that result in a remainder \(r\) upon division by 5, represented as \(x = 5q + r\). Each integer can belong to only one of these sets, defined by a specific remainder.The concept of integer sets is crucial because it allows us to partition the whole set of integers, \(\mathbb{Z}\). A partition of \(\mathbb{Z}\) ensures that every integer fits into exactly one subset without any overlaps. In this scenario, the sets \(\{A_0, A_1, A_2, A_3, A_4\}\) comprehensively cover all integers based on their modular arithmetic properties.

Modular Arithmetic

Modular arithmetic is sometimes referred to as 'clock arithmetic.' It simplifies calculations by working with numbers cyclically, often using remainders as markers for values. Essentially, it's the study of equivalence classes of numbers based on their remainders when divided by a fixed number, called the modulus.
In our exercise, the modulus is 5, and we are interested in how integers behave when divided by this modulus.Modular arithmetic tells us that two numbers are equivalent if they leave the same remainder when divided by the modulus. For instance, 12 and 17 would be equivalent with respect to the modulus 5, as both leave a remainder of 2.This arithmetic forms the basis for the partition of integers into the sets \(\{A_0, A_1, A_2, A_3, A_4\}\). These sets are inherently non-overlapping, which means they cover every integer with respect to their possible remainders when divided by 5. Understanding modular arithmetic allows students to visualize and work with numbers in cycles, providing a unique perspective on number relationships and partitions.