Problem 2 - Advanced Calculus

Question

Show that the set \(\mathrm{A}=\\{\mathrm{x} \in \mathrm{R} \mid 0<\mathrm{x}<1\\}\) is uncountable. Conclude that \(R\) is uncountable.

Step-by-Step Solution

Answer

To show that the set \(A=\{x \in \mathbb{R} \mid 0

Step 1: Understand what "uncountable" means

Uncountable means that the cardinality (size) of the set is larger than the cardinality of any countable set. In other words, it's not possible to list all the elements of an uncountable set in a sequence, such as \(a_1, a_2, a_3, ...\).

Step 2: Start the proof by contradiction for the set \(A\).

Assume that \(A\) is countable. Then there exists a sequence of its elements, i.e., there exist real numbers \(x_1, x_2, x_3, \ldots\) between 0 and 1 such that every real number between 0 and 1 appears in this sequence.

Step 3: Construct a new real number

Consider a new number \(\alpha\) such that its i-th decimal is different from the i-th decimal of \(x_i\) for each \(i\). This can be done by adding 1 to the i-th decimal of \(x_i\) (if this number is less than 8) or by subtracting 1 (if the i-th decimal of \(x_i\) is 9). This way, \(\alpha\) also lies between 0 and 1.

Step 4: Show a contradiction

Since \(\alpha\) is different from every \(x_i\) at digit \(i\), it follows that \(\alpha\) is not in the sequence \((x_i)\). But \(\alpha\) is a real number between 0 and 1, contradicting our initial assumption that every real number between 0 and 1 appears in the sequence. Therefore, \(A\) is uncountable.

Step 5: Extend the argument to the set \(\mathbb{R}\)

Notice that any real number can be written as the sum of an integer and a number in \(A\). If \(\mathbb{R}\) was countable, both sets of integers and \(A\) would be countable (since the union of countable sets is countable). However, in the previous steps, we have shown that \(A\) is not countable. Therefore, \(\mathbb{R}\) can't be countable. We conclude that \(\mathbb{R}\) is uncountable.

Key Concepts

Cardinality of Sets

The concept of cardinality is central to understanding different types of infinities within mathematics. In simple terms, cardinality refers to the size of a set, which can be thought of as the number of elements the set contains. Sets can be finite or infinite, and among infinite sets, some are countable while others are uncountable.

Countable sets are those whose elements can be matched one-to-one with the set of natural numbers. Examples include the set of integers or the set of rational numbers. Even though these sets are infinite, you can establish a sequence that would list all their elements. However, when you deal with an uncountable set, such as the set of real numbers between 0 and 1, you hit a wall. No matter how you try to list them, there will always be numbers that are missing. This characteristic showed us that not all infinities are equal; some are larger—meaning they have a greater cardinality—than others. Cantor's diagonal argument, which is often used in proofs involving uncountability, is one such method that showcases this vast landscape of infinite cardinalities.

Proof by Contradiction

Proof by contradiction is a powerful mathematical technique used to establish the truth of a statement by showing that the opposite assumption leads to a contradiction. In this method, you start by assuming the opposite of what you want to prove. Then, through logical deductions, you arrive at a statement that contradicts either the initial assumption or a well-known fact. This contradiction implies that the initial assumption was false, hence, proving the original statement to be true.

This type of proof is particularly useful when dealing with complex concepts such as uncountability, where direct proof can be challenging. The provided exercise used proof by contradiction to demonstrate the uncountability of a set: by assuming the set was countable and showing this assumption led to an absurdity, the original claim that the set is uncountable was proven beyond doubt.

Real Numbers

Real numbers encompass all the numbers on the continuous number line, which includes both rational numbers (numbers that can be expressed as a quotient of two integers) and irrational numbers (numbers that cannot be expressed as a simple fraction). This vast set is represented by the symbol \(\mathbb{R}\) and encompasses everything from the square root of two to pi to transcendental numbers and beyond.

The real numbers' defining characteristic is their completeness: between any two real numbers, no matter how close they are, there exist infinitely many other real numbers. This density is what makes the set of real numbers uncountable, a concept touched upon in the given exercise. Understanding the nature of real numbers opens up a deeper insight into why certain infinite sets, such as the ones found within \(\mathbb{R}\), are necessarily uncountable and are larger in cardinality than sets like the integers or the natural numbers.

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Problem 1

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Problem 3

Other exercises in this chapter

Problem 1

Show that the set \(\mathrm{Q}\) of rational numbers \(\mathrm{x}\) such that \(0

View solution

Problem 3

Define boundedness and state the property of real numbers concerning least upper bounds (or greatest lower bounds).

View solution

Problem 4

If \(a>0\) and \(b>0\), show that there exists an integer \(\mathrm{n}\) such that na \(>\mathrm{b}\).

View solution

Problem 6

Prove the following give that the sequences \(\left\\{\mathrm{s}_{\mathrm{n}}\right\\}\) and \(\left\\{\mathrm{t}_{\mathrm{n}}\right\\}\) converge to \(\mathrm{

View solution

Key Concepts

Cardinality of Sets

The concept of cardinality is central to understanding different types of infinities within mathematics. In simple terms, cardinality refers to the size of a set, which can be thought of as the number of elements the set contains. Sets can be finite or infinite, and among infinite sets, some are countable while others are uncountable.

Countable sets are those whose elements can be matched one-to-one with the set of natural numbers. Examples include the set of integers or the set of rational numbers. Even though these sets are infinite, you can establish a sequence that would list all their elements. However, when you deal with an uncountable set, such as the set of real numbers between 0 and 1, you hit a wall. No matter how you try to list them, there will always be numbers that are missing. This characteristic showed us that not all infinities are equal; some are larger—meaning they have a greater cardinality—than others. Cantor's diagonal argument, which is often used in proofs involving uncountability, is one such method that showcases this vast landscape of infinite cardinalities.

Proof by Contradiction

Proof by contradiction is a powerful mathematical technique used to establish the truth of a statement by showing that the opposite assumption leads to a contradiction. In this method, you start by assuming the opposite of what you want to prove. Then, through logical deductions, you arrive at a statement that contradicts either the initial assumption or a well-known fact. This contradiction implies that the initial assumption was false, hence, proving the original statement to be true.

This type of proof is particularly useful when dealing with complex concepts such as uncountability, where direct proof can be challenging. The provided exercise used proof by contradiction to demonstrate the uncountability of a set: by assuming the set was countable and showing this assumption led to an absurdity, the original claim that the set is uncountable was proven beyond doubt.

Real Numbers

Real numbers encompass all the numbers on the continuous number line, which includes both rational numbers (numbers that can be expressed as a quotient of two integers) and irrational numbers (numbers that cannot be expressed as a simple fraction). This vast set is represented by the symbol \(\mathbb{R}\) and encompasses everything from the square root of two to pi to transcendental numbers and beyond.

The real numbers' defining characteristic is their completeness: between any two real numbers, no matter how close they are, there exist infinitely many other real numbers. This density is what makes the set of real numbers uncountable, a concept touched upon in the given exercise. Understanding the nature of real numbers opens up a deeper insight into why certain infinite sets, such as the ones found within \(\mathbb{R}\), are necessarily uncountable and are larger in cardinality than sets like the integers or the natural numbers.