Empty Set Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Empty Set.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The empty set, written \emptyset or \{\}, is the unique set containing no elements whatsoever.

Think of an empty box that is still a valid box—it just holds nothing. The empty set plays the same role for sets that zero plays for numbers: it is the identity element for union (A \cup \emptyset = A) and the annihilator for intersection (A \cap \emptyset = \emptyset). It is also a subset of every set, which keeps logical statements about 'all elements of \emptyset' vacuously true.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: There is exactly one empty set — all ways of writing a set with no elements are equal. It is a subset of every set.

Common stuck point: \emptyset \neq \{\emptyset\}. The empty set is different from a set containing the empty set.

Sense of Study hint: Ask yourself: 'Can I name even one element in this set?' If not, the set is empty. Then count: the empty set has 0 elements, but {empty set} has 1.

Worked Examples

Example 1

easy
Determine whether each set is empty: (a) \{x \in \mathbb{R} : x^2 = -1\}, (b) \{x \in \mathbb{Z} : 2 < x < 3\}, (c) \{0\}.

Solution

  1. 1
    (a) x^2 = -1 has no real solution since squares are non-negative. This set is empty: \emptyset.
  2. 2
    (b) There is no integer strictly between 2 and 3. This set is empty: \emptyset.
  3. 3
    (c) \{0\} contains the element 0. It is not empty; it has cardinality 1.

Answer

(a)\;\emptyset,\quad (b)\;\emptyset,\quad (c)\;\text{not empty}
The empty set \emptyset contains no elements at all. A set containing zero (\{0\}) is not empty — 0 is a perfectly valid element.

Example 2

medium
Prove that the empty set \emptyset is a subset of every set A.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Decide which are true: (a) \emptyset = \{0\}, (b) \emptyset \subseteq \{1,2,3\}, (c) |\emptyset| = 0.

Example 2

medium
Let A = \emptyset. Find: (a) A \cup B for any set B, (b) A \cap B for any set B, (c) \mathcal{P}(A) (the power set of A).
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Related Concepts

SetCardinality

Background Knowledge

These ideas may be useful before you work through the harder examples.

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