Proving the Same Cardinality between the Interval [0,1] and the Cartesian Product [0,1] × [0,1]

In elementary set theory, we often encounter the concept of cardinality, which measures the size of a set. Two sets are said to have the same cardinality if there exists a bijection (a one-to-one and onto function) between them. In this article, we will demonstrate that the interval [0,1] and the Cartesian product [0,1] × [0,1] have the same cardinality. This property is significant in topology and analysis and forms a cornerstone in understanding the size of infinite sets.

Step 1: Cardinality of [0,1]

The interval [0,1] is an open interval of real numbers. It is well-known that this set is uncountably infinite, meaning it has the same cardinality as the entire set of real numbers, denoted by mathbb{R}. This is a fundamental result in set theory, often referred to as Cantor's diagonal argument, which shows that the real numbers are uncountable.

Step 2: Cardinality of [0,1] × [0,1]

The Cartesian product [0,1] × [0,1] consists of all ordered pairs (x, y) where x and y are in the interval [0,1]. This product set is also uncountably infinite. The cardinality of this product set is the same as the cardinality of the real numbers, mathbb{R}. To prove this, we need to establish a bijection between [0,1] and [0,1] × [0,1].

Step 3: Constructing the Bijection

To establish a bijection between [0,1] and [0,1] × [0,1], we can use a specific mapping. One common approach is to use a function that interleaves the decimal or binary expansions of numbers in [0,1]. Let’s consider the binary expansion for this purpose:

Choose a number in [0,1]: Let x be a number in [0,1]. We can express x in its binary expansion as:

x 0.a_1 a_2 a_3 ldots

where a_i are the binary digits 0 or 1.

Map to [0,1] × [0,1]: Define a function f: [0,1] to [0,1] times [0,1] by interleaving the digits of x with another number y that we will define. Specifically:

f(x) left(0.a_1 a_3 a_5 ldots, 0.a_2 a_4 a_6 ldotsright)

In this mapping, the first component of the output is formed by taking the first, third, fifth, etc., digits of the binary expansion of x, and the second component takes the second, fourth, sixth, etc., digits.

Step 4: Inverse Function

To show that this function is indeed a bijection, we also need to construct an inverse function g: [0,1] times [0,1] to [0,1]. Given a pair (y, z) in [0,1] × [0,1]:

Let y 0.b_1 b_2 b_3 ldots text{ and } mi z 0.c_1 c_2 c_3 ldots. Define:

g(y, z) 0.b_1 c_1 b_2 c_2 b_3 c_3 ldots

This function constructs a number in [0,1] by interleaving the digits of y and z.

Step 5: Conclusion

Since we have constructed a bijection f from [0,1] to [0,1] × [0,1] and an inverse g, we can conclude that the cardinalities of [0,1] and [0,1] × [0,1] are the same. Therefore, we have shown that:

[0,1] [0,1] × [0,1]

This demonstrates that both sets have the same cardinality and emphasizes the concept of infinite sets in set theory.