Introduction

Cardinality is a term used to describe the size of a set, or the number of elements in a set. It is the most fundamental concept of set theory and is widely used in mathematics, statistics, and computer science. In this article, we will be discussing how to show that two sets have the same cardinality.

The Definition of Set Equality

Two sets are considered equal if they have the same number of elements. This means that if you have two sets, A and B, and A has three elements and B has three elements, then A and B are considered equal. This is known as set equality.

Showing Sets Have Same Cardinality

In order to show that two sets have the same cardinality, you need to prove that they are equal. This can be done by demonstrating that there exists a bijection (a one-to-one correspondence) between the two sets. A bijection is a function that maps each element in one set to exactly one element in the other set.

For example, if you have two sets A and B, and you want to show that they have the same cardinality, you would need to find a bijection between them. To do this, you could map each element in set A to an element in set B. This would establish a one-to-one correspondence between the two sets, proving that they have the same cardinality.

Using the Cantor-Bernstein Theorem

Another way to show that two sets have the same cardinality is to use the Cantor-Bernstein theorem. This theorem states that if there exists an injection from one set to the other, and an injection from the other set to the first, then the two sets have the same cardinality. An injection is a function that maps each element in one set to one element in the other set, and it is not necessarily a bijection.

For example, if you have two sets A and B, and you want to show that they have the same cardinality, you could find an injection from A to B and an injection from B to A. This would prove that the two sets have the same cardinality according to the Cantor-Bernstein theorem.

Using the Diagonalization Process

The last way to show that two sets have the same cardinality is to use the diagonalization process. This process involves creating a new set by taking the elements from the two sets and combining them in a particular way. This new set will have the same cardinality as the two original sets.

For example, if you have two sets A and B, and you want to show that they have the same cardinality, you could create a new set C by taking the elements from A and B and combining them in a particular way. If C has the same number of elements as A and B, then the two sets have the same cardinality.

Conclusion

In conclusion, there are three ways to show that two sets have the same cardinality. These methods are to demonstrate that there exists a bijection between the two sets, to use the Cantor-Bernstein theorem, and to use the diagonalization process. Each of these methods can be used to prove that two sets have the same cardinality.