Problem 34 - Discrete Mathematics with Applications

Question

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

Step-by-Step Solution

Answer

The given set can be rewritten using the listing method as: \( \\{n | n \text { is divisible by } 2\\} = \\{2, 4, 6, 8\\} \)

Step 1: Identify the numbers between 1 and 10

We start by listing all numbers between 1 and 10, excluding 10 itself: \[ 1, 2, 3, 4, 5, 6, 7, 8, 9 \]

Step 2: Check each number for divisibility by 2

Now we'll check each number to see if it's divisible by 2 (i.e., has a remainder of 0 when divided by 2). We can use the rules of divisibility for 2: 1. An even number (last digit is 0, 2, 4, 6, or 8) is divisible by 2. 2. An odd number (last digit is 1, 3, 5, 7, or 9) is not divisible by 2.

Step 3: List the numbers divisible by 2

By checking the numbers 1 to 9, we find that the following numbers are divisible by 2: 2: even 4: even 6: even 8: even So, we can rewrite the given set using the listing method as: \( \\{n | n \text { is divisible by } 2\\} = \\{2, 4, 6, 8\\} \)

Key Concepts

Divisibility

Divisibility is a mathematical concept that determines whether a number can be divided evenly by another. Essentially, if you divide a number by another number and there is no remainder, the first number is said to be divisible by the second. This concept simplifies mathematical calculations and problem-solving because it helps to identify relationships between numbers quickly. For example, divisibility rules, such as those for 2, 3, and 5, allow us to evaluate quickly whether a number meets the criteria. For 2, a number is considered divisible if its last digit is even. By understanding divisibility, you can efficiently classify numbers and work through mathematical problems while minimizing errors.

Even Numbers

Even numbers are integers that can be divided by 2 without leaving a remainder. An easy way to identify even numbers is to look at their last digit: if it is 0, 2, 4, 6, or 8, then the number is even. This identification helps in a variety of mathematical concepts and computations, as even numbers possess unique properties.

• For instance, the sum of two even numbers is always even.
• The product of two even numbers is always even, as well.
• Even numbers also play a role in the creation of patterns, sequences, and solving equations efficiently.

Recognizing and understanding even numbers simplifies tasks such as finding factors, least common multiples, and working with arrays. In our exercise, identifying even numbers was crucial to rewriting the set using the listing method.

Listing Method

The listing method is a straightforward way of displaying the elements of a set. Instead of describing the properties of elements in a set, each element is listed individually. This method is especially useful for sets with a small number of elements, making it easy to view and verify the data. To use the listing method effectively:

• Start by ensuring a clear understanding of the criteria for the set.
• Work through the possible numbers or items to identify which meet the criteria.
• List each qualifying number or item in the set, separated by commas and enclosed in curly braces \(\{ \} \).

In the context of our exercise, the listing method involved evaluating numbers from 1 to 9 for divisibility by 2, then listing those that qualified. This process made it clear and easy to see that numbers 2, 4, 6, and 8 belong to the set \(\{n | n \text{ is divisible by } 2\}\). Thus, it effectively communicates all valid elements of the set without ambiguity.