Problem 22 - 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. \(N\) ______ \(I\)

Step-by-Step Solution

Answer

\(N \subseteq I\)

Step 1: Understand the Set Definitions

To solve the problem, we need to first understand the types of numbers each set represents. The set \(N\) consists of natural numbers, \(I\) consists of integers, \(W\) includes whole numbers, \(Q\) includes rational numbers, \(H\) includes irrational numbers, and \(R\) consists of real numbers. Natural numbers are positive integers starting from 1.

Step 2: Analyze Inclusion Test

Determine if every element in \(N\) is also in \(I\). Natural numbers \(N\) are 1, 2, 3, etc., which are indeed positive integers. Hence, every natural number is also an integer, but not every integer is a natural number (since integers include negative numbers and zero).

Step 3: Conclude the Relationship

Since every element of \(N\) (natural numbers) is found within \(I\) (integers), but \(N\) doesn't cover all elements in \(I\), we establish that \(N\) is a subset of \(I\). Therefore, the statement \(N \subseteq I\) is true.

Key Concepts

Natural Numbers

Natural numbers are the most basic set of numbers you encounter in mathematics. They include all positive integers starting from 1, such as 1, 2, 3, and so on. These numbers are used to count objects and often referred to as the "counting numbers."

Key characteristics of natural numbers include:

• They do not include zero or any negative numbers.
• They are part of the larger set of integers but represent only the positive, non-zero members of that set.
• They are closed under addition and multiplication, meaning that the sum or product of any natural numbers is also a natural number.

Understanding natural numbers is essential in learning basic arithmetic and foundational maths. They serve as a stepping stone to exploring more complex number sets like integers and real numbers.

Integers

Integers are a broader class of numbers that include natural numbers, zero, and negative numbers. Simply put, integers encompass all whole numbers (both positive and negative) without any decimal or fractional parts.

The set of integers can be represented as:

• Positive integers: Similar to natural numbers (e.g., 1, 2, 3,...)
• Zero: A unique integer that is neither positive nor negative
• Negative integers: Opposite counterparts of natural numbers (e.g., -1, -2, -3,...)

Integers are widely used in mathematics for different types of calculations and solving algebraic equations. When we say that natural numbers are a subset of integers, it means that all natural numbers are included within the set of integers, but integers contain more than just the positive numbers. This includes zero and the negative counterparts of natural numbers.

Subset

A subset is an important concept in set theory. When we say that a set A is a subset of set B, it means that all the elements of set A are also elements of set B.

Here's what you need to know about subsets:

• If A is a subset of B, we use the notation \(A \subseteq B\).
• All elements in A are contained in B, but B may have additional elements not found in A.
• If A has no elements not found in B, then A is definitely a subset.

In our specific problem, the natural numbers (N) form a subset of the integers (I), because every natural number is indeed an integer. However, integers include more numbers, such as negative integers and zero, which are not part of the natural numbers set. This understanding helps us confirm that \(N \subseteq I\), illustrating the nested nature of these numerical sets.