Empty Set
High School
Definition
Sometimes known as a null set. It is a set that does not contain any elements within it. Therefore it has a size and cardinality of zero. The notation for empty sets are written as: "{ }" or "∅"
Worked examples
\(\{x \in \mathbb{R} \mid x^2 = -1\} = \emptyset\)
No real number squared gives negative one, so the solution set is empty.
\(\{2, 4, 6\} \cap \{1, 3, 5\} = \emptyset\)
The two sets share no common elements, so their intersection is the empty set.
Common mistakes
- \(\{0\}\) is the empty set → \(\{0\}\) has one element; \(\{\}\) or \(\emptyset\) is empty A set containing zero is not empty — it has cardinality 1.
- \(\emptyset = 0\) → \(\emptyset\) is a set; 0 is a number The empty set is not the same as the number zero — it is a set with size zero.
- Write the empty set as \(\{\emptyset\}\) → Write \(\emptyset\) or \(\{\}\); \(\{\emptyset\}\) has one element Placing the empty-set symbol inside braces creates a set containing the empty set, which has cardinality 1.
Where you'll use it next
You'll use the empty set when solving equations with no solutions, finding set intersections and complements, and later in logic, probability, and advanced topics like topology.
Found in 1 StudyPug lesson
Set Notation: The Foundation of Mathematical Language
10th Grade10thGeometry
Dive into the world of set notation and unlock powerful mathematical concepts. Learn to represent, manipulate, and visualize sets with confidence, enhancing your problem-solving skills across various disciplines.
See also
Reviewed by Pat Cheng, M.Ed. — StudyPug Curriculum Lead · Last updated June 6, 2026