The question asks to identify the correct characteristic of the set of natural numbers, \(N\), with respect to its boundedness within the set of real numbers, \(R\).

Natural Numbers Explained

The set of natural numbers, denoted by \(N\), consists of the positive integers used for counting. It is typically represented as:

• \(N = \{1, 2, 3, 4, \ldots\}\)

Understanding the properties of this set is fundamental in mathematics.

Understanding Boundedness in Real Numbers

To determine if a set like natural numbers is bounded in \(R\), we need to understand the definitions of "bounded below" and "bounded above":

• Bounded Below: A set \(S\) is bounded below if there exists a real number \(m\) (called a lower bound) such that for every element \(x\) in the set \(S\), \(x\) is greater than or equal to \(m\). In mathematical terms, this means that for all \(x \in S\), \(x \ge m\). If no such \(m\) exists, the set is unbounded below.
• Bounded Above: A set \(S\) is bounded above if there exists a real number \(M\) (called an upper bound) such that for every element \(x\) in the set \(S\), \(x\) is less than or equal to \(M\). Mathematically, for all \(x \in S\), \(x \le M\). If no such \(M\) exists, the set is unbounded above.

Analyzing Natural Numbers for Boundedness

Let's apply these definitions to the set of natural numbers, \(N = \{1, 2, 3, \ldots\}\).

Natural Numbers Bounded Below

For the set \(N\), the smallest element is 1. We can clearly find a real number, for instance, \(m = 1\), such that every natural number is greater than or equal to 1. For example, \(1 \ge 1\), \(2 \ge 1\), \(3 \ge 1\), and so on. Any real number less than or equal to 1 (like 0, -10, etc.) would also be a lower bound. Therefore, the set of natural numbers \(N\) is indeed bounded below in \(R\).

Natural Numbers Unbounded Above

Now, let's consider if the set \(N\) is bounded above. This would imply that there is some real number \(M\) that is greater than or equal to every natural number. However, this is not possible. For any real number \(M\) you choose, you can always find a natural number \(n\) that is larger than \(M\). For example, if \(M = 1000\), then \(1001\) is a natural number that is greater than \(M\). This concept is formally supported by the Archimedean property of real numbers, which essentially states that the set of natural numbers is not bounded above. Since we can always find a larger natural number, there is no single real number that can act as an upper bound for the entire set \(N\). Thus, the set of natural numbers \(N\) is unbounded above in \(R\).

Evaluating the Options for Natural Numbers

Let's review the given options based on our analysis of the natural numbers:

• Option 1: "unbounded below in R" - This is incorrect. As established, \(N\) is bounded below by 1.
• Option 2: "bounded above in R" - This is incorrect. \(N\) is unbounded above, meaning there is no upper limit.
• Option 3: "unbounded above in R" - This is correct. The set of natural numbers extends infinitely without an upper limit in the real numbers.
• Option 4: "bounded above and bounded below in R" - This is incorrect. While \(N\) is bounded below, it is not bounded above.

Therefore, the set of natural numbers is unbounded above in the set of real numbers.