Cardinality

What is Cardinality?

Cardinality is just how many elements are in a set.

Notation: $|A|$

This reads: “the cardinality of A” or “the size of A”

Examples

The empty set has cardinality 0.

Cardinality of Union

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

Why subtract the intersection? Elements in both sets get counted twice if you just add.

Example:

$A = \{1, 2, 3\}, \quad B = \{2, 3, 4\}$

• $|A| = 3$
• $|B| = 3$
• $A \cap B = \{2, 3\}$, so $|A \cap B| = 2$

$|A \cup B| = 3 + 3 - 2 = 4$

Check: $A \cup B = \{1, 2, 3, 4\}$ — yes, 4 elements.

Disjoint Sets

If two sets have no overlap:

$A \cap B = \emptyset$

Then:

$|A \cup B| = |A| + |B|$

No need to subtract — nothing is counted twice.

Cardinality of Cartesian Product

$|A \times B| = |A| \times |B|$

Multiply the sizes.

Example:

$A = \{1, 2\}, \quad B = \{a, b, c\}$

$|A \times B| = 2 \times 3 = 6$

Cardinality of Power Set

$|\mathcal{P}(A)| = 2^{|A|}$

A set with $n$ elements has $2^n$ subsets.

Example:

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

$|\mathcal{P}(A)| = 2^3 = 8$

Summary

Cardinality = counting elements.