How big is a set? If it has finitely many elements, the answer is just a number — the cardinality, written |A|. A set with no elements has cardinality 0 (and gets a special name, the empty set). A set with five elements has cardinality 5. A set with infinitely many elements — like the natural numbers \mathbb{N} — has a cardinality too, but it is not an ordinary number; we come back to that at the end.

This page makes cardinality visual: as you conceptually drag elements into a set, the meter climbs; drag them out, it falls; and the empty set is the bottom of the scale.

A meter you can watch

Imagine a bucket labelled A. Drop an apple in — |A| = 1. Drop in another apple of the same kind — it still says |A| = 1, because sets do not count duplicates (an element is either in or not in; repeat mentions do nothing). Drop in a banana — |A| = 2. Drop in an orange — |A| = 3. Take the banana out — |A| = 2. And so on.

A bucket labelled A with four elements and a cardinality meter climbing to four A vertical vertical bar graph on the right shows a meter climbing from zero to a value of four, with tick marks at each integer. On the left, a bucket labelled A contains four distinct shapes — a star, a triangle, a square, and a circle. An arrow from the bucket to the meter indicates the mapping from the set to the count. Below the bucket is a smaller empty bucket labelled empty set with the meter reading zero. A ∅ (empty set) count 0 1 2 3 4 5 |A| = 4 |∅| = 0
A set $A$ with four distinct elements — a star, a triangle, a square, and a circle — registers $|A| = 4$ on the meter. The empty set $\varnothing$ below registers $|A| = 0$. Each integer tick on the meter corresponds to one element in the bucket.

Three rules the meter obeys, which are exactly the rules cardinality obeys:

  1. Adding a new element raises the count by one. |A \cup \{x\}| = |A| + 1 whenever x \notin A.
  2. Adding a duplicate does nothing. If x \in A already, then A \cup \{x\} = A and the count stays the same.
  3. The empty set reads zero. |\varnothing| = 0.

The empty set is not the same as "no set"

Cardinality 0 is a real number. Call it a valid answer. The empty set \varnothing (also written \{\}) is the set with no elements, and it has |\varnothing| = 0. This is different from not having a set at all — the empty set is a perfectly good set, just one with nothing inside.

Why is this worth emphasising? Because later chapters use the empty set constantly — as the result of an intersection with no common elements (A \cap B = \varnothing means A and B are disjoint), as the unique subset with no members, as the answer to a set-builder definition where no element satisfies the condition (\{x \in \mathbb{R} : x^2 = -1\} = \varnothing in the reals).

A common confusion: is \{\varnothing\} the same as \varnothing? No. \{\varnothing\} is a set containing one element (which happens to itself be the empty set), so |\{\varnothing\}| = 1. The container is not empty; it contains an empty box. This distinction matters for the power-set chapter.

Cardinality respects basic operations

The meter behaves predictably when you combine sets. Three important rules, all with memorable pictures:

Union with disjoint sets. If A and B have no common elements, |A \cup B| = |A| + |B|. Two buckets, pour them together, the count adds.

Union with overlap. If A and B have some common elements, the overlap is counted twice if you just add. So the correct formula is inclusion-exclusion:

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

Subtract the overlap once to avoid double-counting. This is the formula behind every "how many students play at least one sport?" question in the chapter.

Difference. |A \setminus B| = |A| - |A \cap B|. What is in A but not in B equals everything in A minus the part shared with B.

Each rule is something you can verify on the meter: construct the sets, count, and check.

Order and labels do not count — elements count

Two sets are equal when they have the same elements, regardless of the order those elements are written. So \{1, 2, 3\} = \{3, 1, 2\} = \{2, 2, 1, 3\} — the first two are clearly the same set listed differently; the third has a duplicate 2 which does not register, so it is the same set too, with cardinality 3.

The cardinality cares only about distinct elements. "Size" of a set is not "number of symbols written between the braces" — it is the number of truly different elements once duplicates are collapsed.

A useful check when counting: rewrite the set so each element appears exactly once, then count. So the cardinality of \{a, b, a, c, b, d, a\} is |\{a, b, c, d\}| = 4.

Infinite cardinality — a preview

What happens with |\mathbb{N}| = |\{1, 2, 3, 4, \ldots\}|? The count never stops. The meter climbs and climbs. Here mathematicians introduce a special symbol — \aleph_0 (aleph-null), read "countably infinite" — to name this size. All sets you can put in a one-to-one correspondence with the natural numbers have this same cardinality, including the integers \mathbb{Z} and the rationals \mathbb{Q}.

And — surprisingly — there are even larger infinite cardinalities. The real numbers \mathbb{R} have a strictly bigger cardinality than \mathbb{N}, even though both are infinite. This is Cantor's theorem, and it is the starting point for the strange, beautiful subject of infinite cardinals. In class 11 you only need |A| \in \{0, 1, 2, 3, \ldots\}, but it is useful to know the meter extends into infinity in more than one way.

The reflex

When a set problem asks for a count, draw the meter. Every distinct element adds one. Duplicates do nothing. The empty set is 0. For unions, use inclusion-exclusion. For differences, subtract the overlap. The meter formalises what cardinality is, and the rules are just statements about how the count changes when you push elements around.

Related: Sets - Introduction · Venn Diagrams from Scratch · Set-Builder Notation Translator · Set Operations