Problem 24 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

\(H \nsubseteq Q\) because irrational numbers are not rational.

Step 1: Understanding Set Definitions

First, recall the definitions of each set. The set \(H\) includes all irrational numbers, which are numbers that cannot be expressed as a simple fraction (i.e., they have non-repeating and non-terminating decimal expansions). The set \(Q\) includes all rational numbers, which can be expressed as the quotient or fraction \(\frac{p}{q}\) of two integers, where \(q eq 0\).

Step 2: Analyze the Relationship

Identify if one set is a subset of the other. Since irrational numbers (\(H\)) cannot be expressed as fractions, they cannot belong to the set of rational numbers (\(Q\)). Therefore, every element in \(H\) does not belong to \(Q\), indicating that \(H\) is not a subset of \(Q\).

Step 3: Determine the Correct Symbol

Since no element of set \(H\) (irrational numbers) can be found in set \(Q\) (rational numbers), we use the symbol \(subseteq\) to denote that \(H\) is not a subset of \(Q\).

Key Concepts

Natural Numbers

Natural numbers are one of the most fundamental concepts in mathematics. They are essentially "counting numbers." These are the numbers you naturally start counting with, such as 1, 2, 3, and so on. Importantly, natural numbers do not include zero or any negative numbers. They are all positive integers.
Natural numbers are denoted by the symbol \(N\), and are part of the larger set of integers.

• Begin at 1, not 0.
• Positive integers only.
• No fractions or decimal numbers.

Natural numbers are used for counting items or ordering them, which makes them indispensable in real life and various mathematical applications.

Irrational Numbers

Irrational numbers are intriguing because they cannot be expressed as a simple fraction of two integers. This means there is no possible way to write an irrational number as \( \frac{p}{q} \), where both \(p\) and \(q\) are whole numbers, and \(q eq 0\).
These numbers have non-repeating, non-terminating decimal expansions, making them unique. A well-known example of an irrational number is \(\pi\), which represents the circumference to the diameter of a circle. Another example is the square root of any prime number, like \(\sqrt{2}\).

• Non-repeating decimals.
• Non-terminating decimals.
• Cannot be expressed as \( \frac{p}{q} \).

Irrational numbers are denoted by the letter \(H\) and form a part of the real numbers, but they distinctly stand apart from rational numbers.

Rational Numbers

Rational numbers can be described as numbers that can be expressed as the ratio of two integers. The typical representation of a rational number is \(\frac{p}{q}\), where \(p\) and \(q\) are integers, and \(q eq 0\).
This means not only integers like 7 and -3 are rational, but any numbers that can be expressed as simple fractions, such as \(\frac{1}{2}\) or \(0.75\) (which equals \(\frac{3}{4}\)).

• Includes both positive and negative numbers.
• Whole numbers and fractions.
• Includes decimal numbers that terminate or repeat.

Rational numbers are labeled with the symbol \(Q\) and are part of the real number system. Together with irrational numbers, they form the complete set of real numbers.

Subset Relations

Understanding subset relations is critical for grasping set theory. A set \(A\) is considered a subset of set \(B\) if every element in \(A\) is also an element of \(B\). The notation for subsets is \(A \subseteq B\), indicating \(A\) is contained in \(B\). If \(A\) is not a subset of \(B\), we write \(A subseteq B\).
In the context of our problem, we assess whether set \(H\) (irrational numbers) is a subset of \(Q\) (rational numbers). Given that irrational numbers cannot be expressed as fractions, there are no common elements between the two. Hence, \(H subseteq Q\) holds true as there is no intersection.

• A subset contains elements all found in another set.
• Uses symbols \(\subseteq\) or \(subseteq\) .
• Helps in categorizing relationships between different sets.

Learning subset relations are key to understanding how different categories in mathematics relate to each other and where they diverge.