Problem 23 - 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. \(Q\)_____ \(H\)

Step-by-Step Solution

Answer

\(Q \nsubseteq H\) since rational and irrational numbers are disjoint.

Step 1: Understand the Set Definitions

Firstly, identify what each set represents: \(N\) represents natural numbers, \(Q\) represents rational numbers, \(W\) represents whole numbers, \(H\) represents irrational numbers, \(I\) represents integers, and \(R\) represents real numbers.

Step 2: Define Members of the Sets

Rational numbers \(Q\) include any number that can be expressed as the quotient of two integers, such as fractions and integers themselves. Irrational numbers \(H\) include numbers that cannot be expressed as simple fractions, such as \(\sqrt{2}\) or \pi\.

Step 3: Consider Properties of Rational and Irrational Numbers

Recognize that rational and irrational numbers are disjoint sets. This means that every number is either rational or irrational, but never both. They do not share any elements.

Step 4: Determine Subset Relationship

Since \(Q\) and \(H\) are disjoint, \(Q\) cannot be a subset of \(H\). Therefore, it should be represented by \(Q subseteq H\).

Key Concepts

Rational Numbers

Rational numbers, denoted by the symbol \(Q\), are numbers that can be expressed as fractions. This means you can write them as the ratio of two integers, say \(a\) and \(b\), where \(b eq 0\). Examples of rational numbers include \(\frac{1}{2}\), \(-3\), and \(5\), because they fit the pattern of fraction \(a/b\). When writing numbers as fractions:

• The numerator \(a\) and denominator \(b\) are integers.
• Decimals that repeat or terminate, like \(0.75\) or \(0.666\cdots\), are also rational. These can be rewritten in fractional form to identify them as rational.

Rational numbers play a key role in mathematics because they include whole numbers, integers, and several everyday values like money measurements. They are closed under addition, subtraction, and multiplication, meaning combining two rational numbers using these operations results in another rational number.

Irrational Numbers

Irrational numbers, noted by the symbol \(H\), are numbers that cannot be written as a simple fraction. They are characterized by non-repeating, non-terminating decimals. Some well-known examples of irrational numbers are:

• \(\pi\), the ratio of the circumference of a circle to its diameter.
• \(\sqrt{2}\), the length of the diagonal of a square with sides of one unit each.

These numbers cannot be perfectly expressed with finite or repeating decimals. The set of irrational numbers fills in the gaps between rational numbers on the number line, providing a complete picture of real numbers. Readers often encounter irrational numbers when dealing with roots of non-perfect squares or transcendental numbers, which are solutions to equations like \(x^{2} - 2 = 0\) that can't be expressed simply as fractions.

Subset Relation

In set theory, understanding subset relations is fundamental. A subset relation, denoted as \(A \subseteq B\), means that every element of set \(A\) is also an element of set \(B\). This category of relationships defines the hierarchy and containment between different sets. When analyzing rational numbers \(Q\) and irrational numbers \(H\), it's crucial to understand that these sets do not overlap at all. They are disjoint sets, meaning they contain no shared elements. Therefore, the statement \(Q \subseteq H\) is false, as rational numbers cannot be found within the set of irrational numbers. It's important to remember that:

• Disjoint sets have no elements in common.
• The real numbers \(R\) are made up of both rational and irrational numbers, yet they remain exclusive in their individual set identities unless specified otherwise.

Knowing how to identify and use subset relations can simplify complex topics in mathematics, allowing for better comprehension of logical connections between different mathematical entities.