Problem 34 - Discrete Mathematics with Applications

Question

In Exercises \(34-37, n\) denotes a positive integer less than \(10 .\) Rewrite each set using the listing method. \(\\{n | n \text { is divisible by } 2\\}\)

Step-by-Step Solution

Answer

The set of positive integers less than 10 and divisible by 2 can be represented using the listing method as: \(\{2,4,6,8\}\).

Step 1: Write down positive integers less than 10

List all positive integers less than 10: 1, 2, 3, 4, 5, 6, 7, 8, 9

Step 2: Determine numbers that are divisible by 2

From the list of positive integers less than 10, identify the numbers that are divisible by 2 (i.e., they have no remainder when divided by 2): 2, 4, 6, 8

Step 3: Rewrite the set using the listing method

Now, rewrite the set as a list of the numbers found in step 2: \(\{2,4,6,8\}\)

Key Concepts

Listing Method

The listing method, also known as the roster or tabular method, is a straightforward yet powerful tool in discrete mathematics for showing the elements of a set in an explicit manner. This technique involves writing out the set members separated by commas and enclosed within curly braces. For instance, if we want to list positive integers less than 5, we would use the listing method to denote the set as \(\{1, 2, 3, 4\}\).

Using the listing method is particularly useful when dealing with a finite number of elements, as it provides a clear and precise depiction of the set's contents. In educational settings, this method encourages students to actively engage with the concept of sets by manually identifying and writing down each member, which aids in better understanding and retention of the material.

Positive Integers

In mathematics, positive integers, also known as natural numbers, are the set of all whole numbers greater than zero. These numbers are crucial in various branches of mathematics and are often denoted as \(\mathbb{N}\) or \(\{1, 2, 3, ...\}\). In the context of the exercise provided, positive integers less than 10 are those that fall within the range from 1 to 9, inclusive.

Understanding positive integers is fundamental for students when exploring other mathematical concepts such as divisibility. Divisibility itself relates to whether one positive integer can be evenly divided by another without leaving a remainder. For example, 4 is divisible by 2 because 4 divided by 2 equals 2, a whole number, with no remainder.

Set Theory

Set theory is a branch of mathematical logic that deals with the collection of objects, known as sets. In basic terms, a set is a grouping of distinct objects considered as a whole. These objects, referred to as elements or members, can be anything: numbers, letters, symbols, or even other sets.

One of the fundamental concepts in set theory is the idea of membership. An object either belongs to a set or does not, and this is determined by the properties that define the set. For example, in the expression \(\{n | n \text{ is divisible by } 2\}\), the symbol '|', read as 'such that', is used to specify the condition that n must satisfy to be included in the set — namely, the divisibility by 2.