Practice Empty Set in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
Quick 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.
Example 1
easyDetermine whether each set is empty: (a) \{x \in \mathbb{R} : x^2 = -1\}, (b) \{x \in \mathbb{Z} : 2 < x < 3\}, (c) \{0\}.
Example 2
mediumProve that the empty set \emptyset is a subset of every set A.
Example 3
easyDecide which are true: (a) \emptyset = \{0\}, (b) \emptyset \subseteq \{1,2,3\}, (c) |\emptyset| = 0.
Example 4
mediumLet 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).