Understanding the Equivalence of Sets of Natural and Odd Natural Numbers

The question whether the set of all odd natural numbers and the set of all natural numbers are equivalent is a fundamental topic in set theory. This article explores the concept of equivalence of sets and provides a one-to-one correspondence, or bijection, between these sets to demonstrate their equal cardinality.

Defining the Sets

We will define the following sets:

The set of all natural numbers (N):
N {1, 2, 3, 4, 5, ...}

The set of all odd natural numbers (O):
O {1, 3, 5, 7, 9, ...}

Establishing the Bijection

To demonstrate that these sets are equivalent, we need to establish a bijection, a function that is both injective (one-to-one) and surjective (onto) between the sets N and O. We will define a function as follows:

Function Analysis

Let's analyze the function by plugging in some values:

For n 1:

For n 2:

For n 3:

For n 4:

And so on...

Properties of the Function

Let's explore the properties of the function:

Injective (One-to-One)

To be injective, different natural numbers must map to different odd natural numbers. If , then:

2n1 - 1 2n2 - 1 > n1 n2

This proves that different natural numbers map to different odd natural numbers.

Surjective (Onto)

To be surjective, for every odd natural number in O, there must exist a natural number in N such that . Given any odd natural number m in O, we can write:

m 2k - 1 for some natural number k.

Thus, for k, we have:

This shows that every odd natural number is mapped by some natural number in N.

Conclusion

Since the function is both injective and surjective, it is a bijection. This proves that the set of all odd natural numbers and the set of all natural numbers have the same cardinality.

Furthermore, discussing the cardinality of the set of all odd numbers (A) and the set of all natural numbers (B), we can note that A and B have the same cardinality. This means that there exists a bijection between these two sets. To extend this concept further, even if we consider the negative odd numbers, the cardinality remains the same, as the function can be adjusted to handle negative values as well.