Cardinality of Sets

The cardinality of a set is the number of elements in it. It tells us how many distinct items are in the set.

If a set $𝐴 =\{2,4,6,8\}$ , then the cardinality of $𝐴$ is 4.

It is denoted by $𝑛(𝐴)$ , meaning “number of elements in set A.”

Example: Let $𝑉=\{𝑎,𝑒,𝑖,𝑜,𝑢\}$ Then, $𝑛(𝑉) = 5$

Q1. What is the cardinality of the set of letters in ‘Mississippi’?

Show solution

The distinct letters in 'Mississippi' are: M, I, S, P. So, the cardinality is $n = 4$ .

Explanation

Distinct Elements - A set always has distinct elements. Meaning, no repetitions allowed!

Key Terms

Term	Meaning
Cardinality	The number of elements in a set
n(A)	Number of elements in set A
n(A ∩ B)	Number of elements common to both A and B
n(A ∪ B)	Number of elements in either A or B or both
$n_o(A)$	Number of elements only in A (i.e., in A but not in B)
Disjoint Sets	Sets with no common elements; $A \cap B = \emptyset$

Disjoint Sets

No common elements:

n(A \cup B) = n(A) + n(B)

Intersecting Sets

Some elements are common:

n(A \cup B) = n(A) + n(B) - n(A \cap B)

Only in A or B

Elements only in A:

n_o(A) = n(A) - n(A \cap B)

Elements only in B:

n_o(B) = n(B) - n(A \cap B)

Universal Set Relationships

If all elements are in A and B only:

n(U) = n(A \cup B)

If U has other elements too:

n(U) = n(A \cup B) + n(\text{outside } A \cup B)