Definition of Countable Sets:

A set $A$ is called countable if:

1. $A$ is finite, or
2. $A$ has the same cardinality as the set of natural numbers $\mathbb{N}$. This means that there exists a bijective function (one-to-one and onto) between $A$ and $\mathbb{N}$. Such a set is also called countably infinite.

In other words, a set is countable if we can list its elements in a sequence $a_1, a_2, a_3, \dots$, even if the process is infinite.

Example of Countable Sets:

• The set of natural numbers $\mathbb{N} = \{1, 2, 3, \dots \}$.
• The set of integers $\mathbb{Z} = \{0, 1, -1, 2, -2, \dots \}$.
• The set of rational numbers $\mathbb{Q}$.

The Set $\{ x \in \mathbb{R} \mid 1 \leq x \leq 2 \}$:

We are interested in determining whether the set $A = \{ x \in \mathbb{R} \mid 1 \leq x \leq 2 \}$ is countable or not. This set represents the interval of real numbers between 1 and 2, including 1 and 2.

Proof that the set is uncountable:

1. The set $A = [1, 2]$ is a subset of the real numbers $\mathbb{R}$, and we know that $\mathbb{R}$ is uncountable (this was proven by Cantor's diagonalization argument).

2. Since the interval $[1, 2]$ is a continuous segment of real numbers, there is no way to list all the real numbers in this interval as a sequence. Even though we might be able to construct sequences of rational numbers within $[1, 2]$, the real numbers in this interval include irrational numbers that cannot be listed in a countable sequence.

3. Cantor’s diagonalization argument can be adapted to show that even within the interval $[1, 2]$, any attempt to enumerate all the real numbers will fail, because we can always construct a new real number within $[1, 2]$ that is not part of any proposed sequence. Hence, $[1, 2]$ is uncountable.

Thus, the set $\{ x \in \mathbb{R} \mid 1 \leq x \leq 2 \}$ is uncountable.

Conclusion:

The set $\{ x \in \mathbb{R} \mid 1 \leq x \leq 2 \}$ is uncountable because it includes all the real numbers in that interval, which cannot be listed or paired with the natural numbers.

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Would you like more details or have any questions about this? Here are five related questions that can expand your understanding:

1. How is Cantor's diagonalization proof used to show the uncountability of real numbers?
2. What is the difference between countable and uncountable sets in terms of cardinality?
3. Can the set of rational numbers within an interval like $[1, 2]$ be considered countable?
4. What are the implications of a set being countable versus uncountable in real-world applications?
5. Can there be a bijection between the set $\mathbb{Q}$ (rational numbers) and the set $\mathbb{R}$ (real numbers)?

Tip: If a set contains a continuous range of real numbers (like any non-degenerate interval), it is typically uncountable.