1. What Does Cardinality Mean?
The cardinality of a set simply tells how many elements are in that set. It is the “size” of the set. Cardinality is written as |A| for a set A.
If a set has 5 elements, its cardinality is 5. If it has no elements, its cardinality is 0.
1.1. Examples
- If \( A = \{2, 4, 6\} \), then |A| = 3.
- If \( B = \{a, b, c, d, e\} \), then |B| = 5.
- If \( C = \emptyset \), then |C| = 0.
2. How to Count Elements in a Set
The method of counting depends on how the set is written. The idea is always the same: count each element exactly once.
2.1. Roster Form
In roster form, elements are listed directly, so count them one by one.
\( D = \{1, 3, 5, 7, 9\} \Rightarrow |D| = 5 \)
2.2. Set-Builder Form
In this case, figure out which elements satisfy the rule, then count them.
Example:
\( E = \{ x \mid x \text{ is an even number less than 12 } \} \)
Elements = 2, 4, 6, 8, 10 → so |E| = 5
3. Cardinality of Finite Sets
A finite set is a set with a limited number of elements. Its cardinality is just the number of elements it contains.
3.1. Example
\( F = \{10, 20, 30, 40\} \Rightarrow |F| = 4 \)
4. Cardinality of Infinite Sets
Some sets go on forever. They have no last element, so their cardinality is infinite. Instead of writing a number, they are described as having infinitely many elements.
4.1. Examples
- Set of natural numbers:
\( |\{1, 2, 3, \ldots\}| = \infty \)
- Set of even numbers:
\( |\{2, 4, 6, 8, \ldots\}| = \infty \)
5. Important Points About Cardinality
Some useful things to remember when working with cardinality:
5.1. Key Ideas
- If two sets have the same number of elements, they have the same cardinality.
- Repeated elements are counted only once because sets never list duplicates.
- The order of elements does not affect cardinality.
- The empty set always has cardinality 0.
6. Practice Conversions
Here are a few forms of sets along with their cardinality to get familiar with counting.
6.1. Examples
- \( G = \{x \mid x \text{ is a letter in the word “LEVEL”}\} \) → Elements: {L, E, V} → |G| = 3
- \( H = \{1, 2, 3, 4, 5, 6\} \) → |H| = 6
- \( J = \{x \mid x \text{ is a multiple of 5}\} \) → Infinite, so |J| = \infty