6.1.7. Set of Real Numbers between 0 and 1

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.

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.

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.

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}.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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).

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