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Chapter 6: Problem 3

List the elements in each of the sets. The set \(I\) of all positive integers no greater than \(6 .\)

Short Answer

Expert verified
The elements in the set $I$ are: \[\{1, 2, 3, 4, 5, 6\}.\]

Step by step solution

01

1. Identify the range of integers

We are given the task to list all positive integers no greater than 6. So, our range of integers is from 1 to 6.
02

2. Listing the integers in the set I

Now that we have a clear understanding of the range, we can list all the positive integers within that range to form set I. Set I will contain the elements: \[\{1, 2, 3, 4, 5, 6\}. \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Positive Integers
Positive integers are the set of all whole numbers greater than zero. These numbers do not include fractions, decimals, or negative values. They start from 1 and go onwards to infinity.
  • Positive integers are often represented by the set notation \( \{1, 2, 3, 4, \ldots\} \).
  • They are used in various mathematical calculations and are fundamental in number theory.

Unlike negative or decimal numbers, positive integers are straightforward and easy to work with. In our exercise, we are focusing on positive integers no greater than 6, which gives us the set \( \{1, 2, 3, 4, 5, 6\} \). Understanding this set helps in solving problems related to counting and order.
Mathematical Sets
Mathematical sets are collections of distinct objects or numbers grouped together in a single entity. Sets are usually denoted using curly braces \( \{ \} \).
  • Each object in a set is called an element.
  • Sets can be finite, meaning they contain a specific number of elements, or infinite.

In set notation, the order of elements does not matter, and each element is unique within the set. For our exercise, the set \( I \) contains the elements \( \{1, 2, 3, 4, 5, 6\} \), which are positive integers. By understanding the concept of sets, students can comprehend collections of numbers or objects in a structured way.
Number Range
Number range refers to a continuous span of numbers between a lowest and highest value. It helps to define the boundaries of a set of numbers.
  • The range includes all numbers from the smallest to the largest number in the set.
  • Number ranges can be defined using inequalities or interval notation.

In our exercise, the number range is defined as all positive integers no greater than 6. This means starting from 1 and listing numbers sequentially up to 6. Addressing number ranges helps in organizing data within certain limits, which can be applied in various mathematical contexts.

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