Set of Real Numbers between 0 and 1
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Introduction to Countable and Uncountable Sets
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Today we're going to explore the concepts of countable vs. uncountable sets. Can anyone share what they think a countable set is?
I think a countable set is one where we can list the elements in a sequence?
Exactly! Countable sets can be enumerated. Now, what about uncountable sets? Does anyone know?
Isn't it a set that can't be listed like the integers?
Correct! Uncountable sets, such as real numbers, cannot be enumerated. Let's dive deeper into this.
Cantor's Diagonalization Argument
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Now we will discuss Cantor's diagonalization argument, which shows that the set of real numbers is uncountable. Can anyone summarize what they think Cantor's theorem states?
Is it about how you can create a new number that’s not in a list?
Yes! By assuming we can list all binary strings, we can create a new binary string that's different at every diagonal position.
So that means our list would always miss at least one string?
Exactly! Therefore, the set of infinite binary strings is uncountable. This demonstrates that not all infinite sets have the same size.
Bijection between {0, 1} and (0, 1)
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Let's explore how to create a bijection between the set {0, 1} and the real numbers between 0 and 1. What do you think a bijection means?
Is it a way to pair elements from both sets one-to-one without leaving any out?
Exactly! So if you take a real number x in (0, 1), its binary representation can be used to form a binary string. Can anyone give me an example?
What about 0.5? Its binary representation is 0.1!
Great! If we chop off the leading '0.' we get the binary string. This demonstrates that there are infinitely many real numbers corresponding to our binary strings!
Introduction & Overview
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Quick Overview
Standard
This section explores the concept of uncountability, focusing on the set of real numbers between 0 and 1. It explains Cantor's diagonalization argument, which demonstrates that this set cannot be enumerated, making it uncountable. The distinction between binary representations of finite and infinite strings is also highlighted.
Detailed
Detailed Summary
In this section, we delve into the concept of uncountability as it relates to the set of all real numbers between 0 and 1, denoted as (0, 1). We begin by recapping the difference between countably infinite and uncountable sets, focusing on Cantor’s diagonalization argument.
Key Concepts
- Countably Infinite vs. Uncountably Infinite: Countably infinite sets can be listed in a sequence, whereas uncountable sets cannot.
- Cantor’s Diagonalization Argument: This proof technique shows that any assumed enumeration of infinite binary strings will always miss at least one string, demonstrating the set's uncountability.
Cantor's Theorem
We affirm that the set of all binary strings of infinite length is uncountable, which utilizes the concept of the diagonal argument to illustrate how a new real number can always be constructed that does not belong to the supposed enumerated set.
We also introduce an important example of a function that creates a bijection between the set {0, 1} and (0, 1), demonstrating that there are as many real numbers between 0 and 1 as there are binary strings of infinite length. This also leads to the conclusion that the set of real numbers (R) is uncountable as well since it includes (0, 1) among other numbers.
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Introduction to the Set of Real Numbers
Chapter 1 of 5
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Chapter Content
So now we know at least one set may be the set {0, 1} which is not countable. Now we will see some other sets as well which are not countable. So what we are going to show here is first the set of real numbers between 0 and 1 but excluding 0 and 1 is uncountable. So the set is denoted by (0, 1) so this is the representation of the set of all real numbers between 0 and 1 excluding 0 and 1.
Detailed Explanation
The set of real numbers between 0 and 1, denoted as (0, 1), consists of all the numbers that lie between 0 and 1, but does not include the endpoints 0 and 1. This section emphasizes that this set is uncountable, indicating that we cannot list all these numbers in a sequence like we can for countable sets.
Examples & Analogies
Think of (0, 1) as a continuous stretch of a road that does not actually start at 0 km nor end at 1 km. No matter how many points you pick along the road, you can always find another point between any two points, which illustrates the endless possibilities within the interval.
Showing Uncountability by Bijection
Chapter 2 of 5
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Chapter Content
So how I am going to show it is uncountable? Well I have already shown that the set {0 , 1} is an uncountable set. I will show you now a bijection between the set {0, 1} and the set of all real numbers between 0 and 1. That will automatically show that; conclude that the set of all real numbers between 0 and 1 has the same cardinality as the set {0, 1}.
Detailed Explanation
To demonstrate the uncountability of the set (0, 1), we can create a one-to-one correspondence (bijection) between the set of binary strings (set {0, 1}) and the real numbers in the interval (0, 1). Since {0, 1} is uncountable, this means (0, 1) is uncountable too, as both sets have the same size in terms of infinity.
Examples & Analogies
Imagine connecting dots on a piece of paper. Each dot represents a binary string, and every point on a line that runs from 0 to 1 represents a real number between 0 and 1. If you can match each dot to a point on this line without missing any, it shows that both are just as 'crowded' with points, hence they are both uncountable.
Understanding the Bijection Function
Chapter 3 of 5
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Chapter Content
So what is the bijection? Bijection is very simple. So if I take any x, any real number between 0 and 1 excluding 0 and 1 that will have a binary representation. So let the binary representation of that real number be 0.y where y is a binary string. So what will be the function f? The function f(x) will be y, that means I will just chop off the 0 here and the point here and just I will focus binary representation that means the bits in the representation y and that will be the mapping of x as per the function.
Detailed Explanation
The bijection function f takes a real number x in the interval (0, 1) and translates it into its binary representation by ignoring the leading '0.' This means that every real number can correspond directly to a binary string, thus confirming that (0, 1) has the same cardinality as the set of all binary strings.
Examples & Analogies
Think of this as a translator of languages. If you have a word in English (the real number), the translator ignores any extra letters or punctuation (the leading '0' and the decimal point) and translates it into a concise form in another language (the binary representation). The essence remains the same in both languages, showing equality in meaning despite different forms.
Exploration of Binary Representations
Chapter 4 of 5
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Chapter Content
Now if I consider the set of real numbers, this set R denotes the set of real numbers then it contains the subset (0, 1) it also includes all the real number between 0 and 1 also. And since (0, 1) the set of all real numbers between 0 and 1 is uncountable and remember we had argued that, if we had a set with a subset which is uncountable then the whole super set will also be uncountable.
Detailed Explanation
The set of all real numbers R includes not only the numbers between 0 and 1 but also integers, rational numbers, and irrational numbers. Since we established that the subset (0, 1) is uncountable, we can conclude that the entire set of real numbers R must also be uncountable. This is because any set containing an uncountable subset must itself be uncountable.
Examples & Analogies
Consider a big city (the set of all real numbers) that contains various neighborhoods (subsets). If one neighborhood is so dense that you can't possibly count all its houses (an uncountable set), it makes sense that the entire city, which includes this neighborhood, also has an uncountable amount of places to live.
The Role of Irrational Numbers
Chapter 5 of 5
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Chapter Content
And intuitively the main reason that the set of real numbers is uncountable is that it has irrational numbers as well which we cannot enumerate out.
Detailed Explanation
Irrational numbers, such as the square root of 2 or pi, do not have terminating or repeating decimal representations. This characteristic makes it impossible to list them in a countable manner with the other numbers. Thus, the presence of these irrational numbers solidifies the argument that the set of real numbers is uncountable.
Examples & Analogies
Imagine a pie divided into slices. Rational numbers are like those slices: you can count them easily. But then think of those tiny crumbs that remain after slicing the pie - those are the irrational numbers. You can't count them because they take on endless, non-repeating forms, making them elusive and indicating the uncountable nature of the entire pie (the set of real numbers).
Key Concepts
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Countably Infinite vs. Uncountably Infinite: Countably infinite sets can be listed in a sequence, whereas uncountable sets cannot.
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Cantor’s Diagonalization Argument: This proof technique shows that any assumed enumeration of infinite binary strings will always miss at least one string, demonstrating the set's uncountability.
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Cantor's Theorem
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We affirm that the set of all binary strings of infinite length is uncountable, which utilizes the concept of the diagonal argument to illustrate how a new real number can always be constructed that does not belong to the supposed enumerated set.
-
We also introduce an important example of a function that creates a bijection between the set {0, 1} and (0, 1), demonstrating that there are as many real numbers between 0 and 1 as there are binary strings of infinite length. This also leads to the conclusion that the set of real numbers (R) is uncountable as well since it includes (0, 1) among other numbers.
Examples & Applications
Example of Cantor's diagonalization: Given a sequence of binary strings, the diagonal string constructed from flipping bits demonstrates that the set is uncountable.
Bijection example: Mapping each real number in (0, 1) to its unique binary representation corresponds to the infinite binary strings.
Memory Aids
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Rhymes
Countable to list, a finite twist; Uncountable will hide, a secret we can't bide.
Stories
Imagine a librarian with infinite books, but some books are always missing. This librarian shows how infinite encompasses more than we perceive, just like the uncountable sets.
Memory Tools
C.B.U - Countably Infinite, Bijection, Uncountable - remember the flow of sets!
Acronyms
CUB - Countable, Uncountable, Bijection, helps to recall key concepts.
Flash Cards
Glossary
- Countably Infinite Set
A set that can be listed in a sequence, allowing for enumeration.
- Uncountable Set
A set that cannot be listed or enumerated.
- Cantor's Diagonalization Argument
A method of proving the uncountability of sets by demonstrating that any assumed enumeration will miss at least one element.
- Bijection
A one-to-one correspondence between the elements of two sets.
Reference links
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