Q.State True or False: $\mathbb{Q} \cup \mathbb{Z} = \mathbb{Q}$, where $\mathbb{Q}$ is the set of rational numbers and $\mathbb{Z}$ is the set of integers.

The key idea here is the definition of set union and the subset relationship between the set of integers and the set of rational numbers.

Every integer $n$ can be expressed as a fraction $\frac{n}{1}$, where $n$ is an integer and $1$ is a non-zero integer. This fits the definition of a rational number. Therefore, the set of integers $\mathbb{Z}$ is a subset of the set of rational numbers $\mathbb{Q}$ (i.e., $\mathbb{Z} \subseteq \mathbb{Q}$). …

Concept and Intuition

This question tests our understanding of fundamental number sets and basic set operations, specifically the union of sets.

• Integers ($\mathbb{Z}$): These are whole numbers, both positive and negative, including zero. Examples: $\dots, -3, -2, -1, 0, 1, 2, 3, \dots$.
• Rational Numbers ($\mathbb{Q}$): These are numbers that can be expressed as a fraction $\frac{p}{q}$, where $p$ and $q$ are integers, and $q$ is not zero. Examples: $1/2, -3/4, 5, 0, -2/1$.

The key to solving this problem lies in understanding the relationship between $\mathbb{Z}$ and $\mathbb{Q}$. We need to determine if one set is contained within the other.

The union of two sets, $A \cup B$, is a new set containing all elements that are in $A$, or in $B$, or in both. If one set is entirely contained within another (i.e., it's a subset), then their union simplifies significantly. For instance, if $A$ is a subset of $B$ ($A \subseteq B$), then every element of $A$ is already in $B$. Therefore, when we combine all elements from $A$ and $B$, we simply get all the elements of $B$.

Step-by-Step Solution

1. Define the sets involved:

   • The set of integers, $\mathbb{Z}$, is defined as $\mathbb{Z} = \{\dots, -2, -1, 0, 1, 2, \dots\}$.
   • The set of rational numbers, $\mathbb{Q}$, is defined as $\mathbb{Q} = \left\{ \frac{p}{q} \mid p \in \mathbb{Z}, q \in \mathbb{Z}, q \neq 0 \right\}$.

2. Examine the relationship between $\mathbb{Z}$ and $\mathbb{Q}$:

   We need to determine if every integer is also a rational number.

   Consider any arbitrary integer $n \in \mathbb{Z}$.

   Can we express $n$ in the form $\frac{p}{q}$ where $p, q \in \mathbb{Z}$ and $q \neq 0$?

   Yes, we can write any integer $n$ as $\frac{n}{1}$.

   Here, $p = n$ (which is an integer) and $q = 1$ (which is an integer and not zero).

   Therefore, every integer $n$ can be expressed as a rational number $\frac{n}{1}$. …