Problem 35 - Discrete Mathematics with Applications

Question

In Exercises \(34-37, n\) denotes a positive integer less than \(10 .\) Rewrite each set using the listing method. \(\\{n | n \text { is divisible by } 3\\}\)

Step-by-Step Solution

Answer

The given set can be rewritten as \( \{3,6,9\} \).

Step 1: Identify the Divisible Numbers

We need to find the positive integers less than 10 that are divisible by 3. Let's list them down: - 3 is divisible by 3, since (3 / 3) = 1 - 6 is divisible by 3, since (6 / 3) = 2 - 9 is divisible by 3, since (9 / 3) = 3

Step 2: Rewrite the Set Using Listing Method

Now that we found the elements that meet the given condition, we can rewrite the set using the listing method. The set will be: {3,6,9}

Key Concepts

Listing Method

The listing method, also known as the roster method, is a way to represent sets by explicitly listing each individual element. This method is straightforward and works best when dealing with sets containing a small and finite number of elements. To use the listing method effectively, you need to identify all the elements that belong to the set based on the conditions specified, and then write them within curly brackets.
For instance, if we are asked to create a set of positive integers less than 10 that are divisible by 3, we first identify the integers—such as 3, 6, and 9—that meet this criterion. Then, using the listing method, we represent this set as \(\{3, 6, 9\}\).
The key aspect of this method is clarity and simplicity in representation, which makes it easy to understand and verify. It is particularly useful in classroom settings and introductory discrete mathematics problems, where the sets to be listed are relatively small.
Remember that each element of the set should be unique, as sets do not contain duplicates.

Divisibility Rules

Divisibility rules provide an easy way to determine when one number is divisible by another, without requiring long division. For a number to be divisible by 3, the sum of its digits must be divisible by 3.
Let’s see how these rules apply to our example of positive integers less than 10. To check these integers, we apply the divisibility rule for 3:

• The number 3 itself is divisible by 3, since 3 divided by 3 equals 1.
• Similarly, 6 is divisible by 3, because 6 divided by 3 equals 2.
• And 9 is divisible by 3, with 9 divided by 3 equaling 3.

These rules help in quickly spotting numbers that belong to specific sets without lengthy calculations. They are handy tools in mathematics that ease the process of solving problems involving divisibility, particularly in discrete mathematics where such concepts frequently appear.

Set Notation

Set notation is the language used to denote collections of objects or numbers that satisfy a certain condition. This notation is a fundamental part of mathematics, particularly discrete mathematics.
In set-builder notation, a set is described using a property that its members must satisfy. In the given exercise, this is initially expressed as \(\{n \ | \ n \text{ is divisible by } 3\}\), meaning the set contains all numbers \(n\) which meet the divisibility condition.
Set notation also encompasses the listing method for displaying identified set members explicitly in curly brackets, such as \(\{3, 6, 9\}\). This notation helps streamline communication of mathematical concepts and is crucial in forming precise logical statements.

• The curly brackets \(\{\}\) are used to denote a set.
• The vertical bar \(|\) in set-builder notation means "such that."
• All members within curly brackets should be separated by commas in the listing method.

Utilizing set notation correctly allows for efficient and effective problem-solving in mathematics.