Problem 34 - Intermediate Algebra

Question

Determine whether each statement is true or false. $$ \mathbb{Q} \nsubseteq \mathbb{Z} $$

Step-by-Step Solution

Answer

The statement \( \mathbb{Q} \nsubseteq \mathbb{Z} \) is true because rational numbers are not all integers.

Step 1: Understanding Sets

bQ is the set of all rational numbers, which can be expressed as fractions \( \frac{a}{b} \) where \( a \) and \( b \) are integers, and \( b eq 0 \).
7bZ is the set of all integers, including negative integers, zero, and positive integers.

Step 2: Interpreting the Statement

The statement \( \mathbb{Q} subseteq \mathbb{Z} \) is asking if \( \mathbb{Q} \), the set of all rational numbers, is not a subset of \( \mathbb{Z} \), the set of all integers. In other words, does there exist at least one rational number that is not an integer?

Step 3: Checking Rationals vs Integers

Rational numbers include numbers like \( \frac{1}{2} \), \( \frac{3}{4} \), etc., which are not integers. Integers must be whole numbers like -3, 0, 4, etc. Since fractions like \( \frac{1}{2} \) exist, which are not integers, \( \mathbb{Q} \) cannot be entirely contained within \( \mathbb{Z} \).

Step 4: Conclusion

Because there exist rational numbers that are not integers, the set of rational numbers \( \mathbb{Q} \) is not a subset of the set of integers \( \mathbb{Z} \). Thus, the statement \( \mathbb{Q} subseteq \mathbb{Z} \) is true.

Key Concepts

Rational Numbers

Rational numbers play a crucial role in algebra and everyday mathematics. They are numbers that can be expressed as the quotient or fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers, and \( b eq 0 \). This means that rational numbers encompass a vast set of values including:

• Whole numbers (for example, 5 can be written as \( \frac{5}{1} \))
• Fractions (such as \( \frac{3}{4} \) or \( \frac{1}{2} \))
• Negative fractions (like \( -\frac{7}{8} \))

A key characteristic of rational numbers is that they can be either positive, negative, or zero. Importantly, every integer is a rational number, since any integer \( n \) can be written as \( \frac{n}{1} \). However, rational numbers are broader than integers and include non-whole numbers as well.

Integers

Integers are like the backbone of number systems, consisting of positive numbers, negative numbers, and zero. They do not include fractions or decimals. The set of integers is denoted by \( \mathbb{Z} \), and it can be represented as:

• \( \ldots, -3, -2, -1, 0, 1, 2, 3, \ldots \)

Integers are crucial because they form the set from which we build other, more complex numbers like rational and irrational numbers. They also help in counting, ordering, and performing arithmetic operations like addition, subtraction, and multiplication.Every integer is a rational number, as mentioned earlier, because it can be expressed in the form \( \frac{n}{1} \). However, not all rational numbers are integers.

Subsets

A subset is a set in which all elements are contained within another set. For example, if we have two sets \( A \) and \( B \), set \( A \) is a subset of set \( B \) if every element in \( A \) is also in \( B \). This is written as \( A \subseteq B \).In our exercise, the question is whether the set of rational numbers \( \mathbb{Q} \) is a subset of the set of integers \( \mathbb{Z} \). For \( \mathbb{Q} \) to be a subset of \( \mathbb{Z} \), every rational number would need to also be an integer, but rational numbers include fractions that are not integers. Thus, \( \mathbb{Q} \) is not a subset of \( \mathbb{Z} \), making the statement \( \mathbb{Q} subseteq \mathbb{Z} \) true.Understanding subsets helps us comprehend how different groups of numbers interact, emphasizing the relation between types of numbers and highlighting that while integers form part of rational numbers, not all rational numbers fit into the integer set.