No, infinite sets do not always have equal cardinality.
While it might seem counterintuitive, different infinite sets can have different "sizes" or cardinalities. The concept of cardinality refers to the number of elements in a set. For finite sets, this is straightforward counting. However, for infinite sets, the concept is more nuanced and involves the possibility of a one-to-one correspondence (bijection) between the elements of two sets.
Understanding Cardinality
- Finite Sets: Two finite sets have the same cardinality if you can count the same number of elements in each.
- Infinite Sets: Two infinite sets have the same cardinality if you can create a one-to-one correspondence between their elements. This means for every element in the first set, there's a unique element in the second set, and vice-versa.
Countable vs. Uncountable Infinity
Infinite sets are classified into countable and uncountable sets.
- Countable Sets: A set is countable if its elements can be put into a one-to-one correspondence with the set of natural numbers (1, 2, 3,...). Examples of countable sets include:
- Natural Numbers (N): By definition.
- Integers (Z): You can create a sequence like 0, 1, -1, 2, -2, 3, -3,... that maps each integer to a natural number.
- Rational Numbers (Q): Although rational numbers appear "denser" than integers, they can also be arranged in a way that allows a one-to-one correspondence with the natural numbers.
- Uncountable Sets: A set is uncountable if it's infinite but cannot be put into a one-to-one correspondence with the natural numbers. A classic example is:
- Real Numbers (R): Georg Cantor famously proved that the set of real numbers between 0 and 1 is uncountable using a diagonalization argument. This proof shows that no matter how you try to list all real numbers between 0 and 1, you can always construct a new real number that's not on the list. Since the real numbers between 0 and 1 are uncountable, the entire set of real numbers is also uncountable.
Examples Illustrating Different Cardinalities
| Set | Cardinality | Countable/Uncountable | Explanation |
|---|---|---|---|
| Natural Numbers (N) | Aleph-null (ℵ₀) | Countable | The "smallest" infinite cardinality. |
| Integers (Z) | Aleph-null (ℵ₀) | Countable | Can be put into a one-to-one correspondence with natural numbers (e.g., 0, 1, -1, 2, -2,...). |
| Rational Numbers (Q) | Aleph-null (ℵ₀) | Countable | Though dense, rational numbers can be systematically listed and paired with natural numbers. |
| Real Numbers (R) | Cardinality of the Continuum (c or 2ℵ₀) | Uncountable | Cantor's diagonalization argument proves that there's no bijection between the real numbers and the natural numbers. The cardinality of the real numbers is strictly greater than Aleph-null. |
Since the real numbers (R) cannot be put into a one-to-one correspondence with the natural numbers (N), the cardinality of R is greater than the cardinality of N. Therefore, infinite sets can indeed have different cardinalities.
