Problem 37 - Discrete Mathematics with Applications

Question

\(n\) denotes a positive integer less than \(10 .\) Rewrite each set using the listing method. \(\\{n | n \text { is divisible by } 2 \text { or } 3\\}\)

Step-by-Step Solution

Answer

The short version of the answer is: \[\{2, 3, 4, 6, 8, 9\}\]

Step 1: Identify numbers divisible by 2

To find the numbers divisible by 2, we need to check all positive integers less than 10 and see if they are divisible by 2 without leaving a remainder. The numbers are: 2, 4, 6, and 8.

Step 2: Identify numbers divisible by 3

To find the numbers divisible by 3, we need to check all positive integers less than 10 and see if they are divisible by 3 without leaving a remainder. The numbers are: 3, 6, and 9.

Step 3: Combine both lists and eliminate duplicates

Now that we have both lists of numbers divisible by 2 and 3, we need to combine them into one list and eliminate any duplicates (because a number can be divisible by both 2 and 3). The combined list looks like this: 2, 3, 4, 6, 8, and 9.

Step 4: Rewrite using the listing method

Finally, we will rewrite the set using the listing method, as required by the problem. The set is written as: \[\{2, 3, 4, 6, 8, 9\}\]

Key Concepts

Divisibility

Divisibility in mathematics refers to the ability of one number to be divided by another without leaving any remainder. For instance, a number is divisible by 2 if it can be divided into two equal whole parts. Similarly, a number is divisible by 3 if the sum of its digits is divisible by 3.
When solving problems related to divisibility, you often need to test numbers to see if they fit criteria like being even for divisibility by 2, or if the sum of a number's digits is complete, indicating divisibility by 3.

• Even numbers are always divisible by 2. Examples: 2, 4, 6.
• If the sum of a number's digits is divisible by 3, the number itself is also divisible by 3. Example: 6 (6), 9 (9).

Understanding divisibility is crucial in mathematical computations because it reduces larger problems into simpler parts.
It helps identify factors and is used in simplifying fractions, solving equations, and finding least and greatest common divisors.

Positive Integers

Positive integers are whole numbers greater than zero, extending from 1 to infinity. They do not include fractions, decimals, or negative numbers. For this exercise, we focus on positive integers less than 10. These are: 1, 2, 3, 4, 5, 6, 7, 8, and 9.
When tackling problems that involve positive integers, it’s important to remember that every number is simple and distinct.

• They are the building blocks of mathematics and often serve as the simplest form of numeric data.
• They represent discrete units like people, objects, or items in a set.

In set theory and other mathematical branches, understanding positive integers is essential for creating sets, performing basic arithmetic operations, and solving problems that require non-negative solutions.

Listing Method

The listing method is a way of writing a set by enumerating its elements within curly braces. It's a straightforward way to convey the members of a set clearly and concisely. For instance, the set of even numbers less than 10 is listed as \( \{2, 4, 6, 8\} \).
When using the listing method, you ensure each item is distinct, and typically, the elements are listed in ascending order starting from the smallest.

• It's helpful for clearly displaying sets of numbers without ambiguity.
• Reduces complex notation to simple, easy-to-read forms.

In the original problem, using the listing method, you are asked to write out all positive integers below 10 that are divisible by 2 or 3. After determining these numbers through the rules of divisibility, you combine them into a single set, ensuring no duplicates, such as \( \{2, 3, 4, 6, 8, 9\} \). This method helps students visualize the solution and strengthens their grasp of set theory concepts.