Cardinality Formula

The cardinality of a finite set is the number of distinct elements it contains, written |A| — it measures the size of the set without regard to element.

The Formula

|A \cup B| = |A| + |B| - |A \cap B|

(inclusion-exclusion principle)

When to use: Cardinality answers "how many?" — count each distinct element once and you have the cardinality.

Quick Example

|\{a, b, c\}| = 3

|\emptyset| = 0

|\{\{1, 2\}, 3\}| = 2

— there are two distinct elements.

Notation

|A|

n(A)

What This Formula Means

The cardinality of a finite set is the number of distinct elements it contains, written $|A|$ — it measures the size of the set without regard to element order or identity.

Cardinality answers "how many?" — count each distinct element once and you have the cardinality.

Formal View

|A \cup B| = |A| + |B| - |A \cap B|

(inclusion-exclusion);

|A| = n \Leftrightarrow \exists

a bijection

f : A \to \{1, 2, \ldots, n\}

Worked Examples

Example 1

easy

Find the cardinality of: (a)

A = \{2, 4, 6, 8, 10\}

, (b)

B = \{x \in \mathbb{N} : x \le 0\}

, (c)

C = \{\{1,2\}, 3, \{4\}\}

Answer

|A|=5,\quad |B|=1,\quad |C|=3

First step

(a) Count the distinct elements:

2, 4, 6, 8, 10

— five elements, so

|A| = 5

Full solution

2
(b) The only natural number $\le 0$ is $0$ (assuming $0 \in \mathbb{N}$ ). So $B = \{0\}$ and $|B| = 1$ .
3
(c) $C$ has three elements: the set $\{1,2\}$ , the number $3$ , and the set $\{4\}$ . So $|C| = 3$ .

Cardinality counts distinct top-level elements. When a set contains other sets as elements, each sub-set counts as one element regardless of its own size.

Example 2

medium

Let

A = \{1,2,3,4\}

and

B = \{3,4,5,6\}

. Verify the formula

|A \cup B| = |A| + |B| - |A \cap B|

Example 3

medium

At a party of

60

35

like jazz,

40

like rock,

20

like both. How many like at least one genre?

Common Mistakes

Counting a repeated listing twice, like $|\{a, a, b\}| = 3$ — distinct elements only, so it is $2$ .
Using $|A| + |B|$ for the union when sets overlap — subtract $|A \cap B|$ via inclusion-exclusion.
Confusing cardinality with the number of subsets — elements count linearly, subsets count as $2^{|A|}$ .

Why This Formula Matters

Cardinality turns sets into counting tools — it underlies probability ( $\frac{|E|}{|S|}$ ), inclusion-exclusion, and combinatorics. A student who counts duplicates, or who adds $|A| + |B|$ without subtracting the overlap, overcounts in every 'how many in either group' problem. Recognizing it by "Am I counting how many distinct elements a set has, each once?" — rather than by familiar numbers — is what lets a student tell it apart from sum of two cardinalities and number of subsets (power set size) and element in a mixed problem set.

Frequently Asked Questions

What is the Cardinality formula?

The cardinality of a finite set is the number of distinct elements it contains, written $|A|$ — it measures the size of the set without regard to element order or identity.

How do you use the Cardinality formula?

Cardinality answers "how many?" — count each distinct element once and you have the cardinality.

What do the symbols mean in the Cardinality formula?

$|A|$ or $n(A)$

Why is the Cardinality formula important in Math?

What do students get wrong about Cardinality?

The procedure for cardinality is the easy part; the trap is counting a repeated listing twice, like $|\{a, a, b\}| = 3$ . Asking "Am I counting how many distinct elements a set has, each once?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Cardinality formula?

Before studying the Cardinality formula, you should understand: set, element.

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Related Concepts

Set Element

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Set Formula Element Formula