Problem 20 - Intermediate Algebra

Question

Use the following set designations. \(N=\\{x \mid x\) is a natural number \(\\}\) \(Q=\\{x \mid x\) is a rational number \(\\}\) \(W=\\{x \mid x\) is a whole number \(\\}\) \(H=\\{x \mid x\) is an irrational number \(\\}\) \(I=\\{x \mid x\) is an integer \(\\}\) \(R=\\{x \mid x\) is a real number \(\\}\) Place \(\subseteq\) or \(\nsubseteq\) in each blank to make a true statement. \(N\) _________ \(R\)

Step-by-Step Solution

Answer

\(N \subseteq R\)

Step 1: Understand Natural Numbers (N)

Natural numbers (N) are the set of all positive whole numbers starting from 1, such as 1, 2, 3, 4, etc. They do not include zero or any negative numbers.

Step 2: Understand Real Numbers (R)

Real numbers (R) include all rational and irrational numbers. This means they encompass every number that can be plotted on a number line, including positive and negative numbers, zero, fractions, and irrational numbers.

Step 3: Establish the Relationship (N and R)

Since the set of natural numbers (N) is composed of positive integers, and these are included in the set of real numbers (R), we can say that every natural number is also a real number.

Step 4: Determine the Set Relationship

Based on the definitions, we conclude that the set of natural numbers (N) is a subset of the set of real numbers (R) because all elements of N are also elements of R. Thus, we place the subset symbol: \(N \subseteq R\).

Key Concepts

Natural Numbers

Natural numbers are one of the most basic sets of numbers in mathematics. They consist of all positive integers starting from 1 and going on indefinitely. This includes numbers like 1, 2, 3, and so on. Often, natural numbers are denoted by the symbol \(N\).
What makes natural numbers unique is that they are the numbers we naturally use in counting things. They are the first numbers that children learn, and they form the basis of other number systems.
It is important to note that natural numbers do not include zero or any negative numbers. This provides a clear distinction between natural numbers and other sets such as whole numbers or integers. Understanding this distinction helps in grasping more advanced mathematical concepts.

Real Numbers

Real numbers include a vast array of numbers that can be found on the number line. They are denoted by the symbol \(R\), and they comprise both rational and irrational numbers.

• Rational numbers are numbers that can be expressed as a fraction of two integers, such as \( \frac{1}{2} \) or \( 4 \). This means they can be positive, negative, or zero.
• Irrational numbers, on the other hand, cannot be expressed as a fraction of integers. Their decimal values are non-repeating and non-terminating, examples include \( \sqrt{2} \) and \( \pi \).

With these components, the set of real numbers covers every possible position on the number line, encapsulating integers, decimals, and other more complex numbers. Real numbers are instrumental because they allow mathematicians and scientists to solve a wide range of problems, from simple calculations to complex equations in physics.

Subset Symbol

In set theory, one crucial concept is the idea of subsets, which is often represented by the subset symbol \( \subseteq \). When we say that a set \(A\) is a subset of set \(B\), it means all elements of \(A\) are also contained within \(B\).
This is an important property because it helps to establish relationships between different sets of numbers or objects. For instance, the relationship \( N \subseteq R \) signifies that every natural number is also a real number.

• The subset symbol \( \subseteq \) is inclusive, meaning it allows for the possibility that the two sets could actually be identical.
• If a set is not a subset of another, it is represented by the symbol \( subseteq \).

Using the subset symbol effectively helps clarify the organization of different number sets, making it easier to understand their connections and hierarchy in set theory. This concept is fundamental in mathematics, as it lays the groundwork for more complex set relations and applications.