Cardinality of a Set refers to the number of elements in a set. It is a measure of the "size" of the set, which can be finite or infinite. If A is a set, the cardinality of set A is denoted as ∣A∣. Based on cardinality or one-to-one correspondence with the set of natural numbers, we can classify the sets into two categories:
- Countable Sets
- Uncountable Sets
Countable Sets
A countable set is a set whose elements can either be put into a one-to-one correspondence with the set of natural numbers (i.e., the set is countably infinite) or whose elements can be counted in a finite amount (i.e., the set is finite).
A set A is countable if:
- Finite Set: A has a finite number of elements. In this case, the set's cardinality ∣A∣ is a non-negative integer.
- Countably Infinite Set: There exists a bijection (one-to-one and onto function) between A and the set of natural numbers N. This means each element of A can be paired with a unique natural number. In this case, the set is said to have a cardinality of ℵ0 (the smallest type of infinity).
Examples of Countable Sets
Some examples of countable sets are:
| Set | Example |
|---|---|
| Natural Numbers | {1, 2, 3, 4, 5, . . .} |
| Integers | {. . . , −3, −2, −1, 0, 1, 2, 3, . . .} |
| Rational Numbers | {p/q ∣ p, q ∈ Z, q ≠ 0} |
| Prime Numbers | {2, 3, 5, 7, 11, 13, 17, . . . } |
| Even Numbers | {2, 4, 6, 8, 10, . . . } |
{1, 3, 5, 7, 9, 11, . . .} |
Note: All finite sets are countable.
Uncountable Sets
An uncountable set is a set that cannot be put into a one-to-one correspondence with the set of natural numbers.
In other words, it is a set that has a larger cardinality than the set of all natural numbers, meaning it cannot be enumerated or counted in a sequential manner.
Note: Uncountable sets are infinite, but they are "larger" than countably infinite sets (like the set of natural numbers).
Examples of Uncountable Sets
Some examples of uncountable sets includes:
- Real Numbers (R)
- Power set of Natural Numbers
- Points on a Line Segment
- Cantor Set
- Complex Numbers
Note: The cardinality of an uncountable set is often denoted as
Countable vs Uncountable Sets
| Countable Sets | Uncountable Sets |
|---|---|
| A set that can be put into a one-to-one correspondence with the natural numbers (N). | A set that cannot be put into a one-to-one correspondence with the natural numbers. |
| The cardinality is either finite or countably infinite (ℵ0). | The cardinality is greater than that of any countable set, often denoted as |
| Natural numbers (N), integers (Z), rational numbers (Q). | Real numbers (R), power set of natural numbers, points on a line segment. |
| Can be listed in a sequence (e.g., a1, a2, a3, . . .). | Cannot be fully enumerated in a sequence. |
| Any subset of a countable set is either countable or finite. | A subset of an uncountable set can be countable or uncountable. |