Empty Set Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

easy
Determine whether each set is empty: (a) {x∈R:x2=−1}\{x \in \mathbb{R} : x^2 = -1\}, (b) {x∈Z:2<x<3}\{x \in \mathbb{Z} : 2 < x < 3\}, (c) {0}\{0\}.

Solution

  1. 1
    (a) x2=−1x^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}\{0\} contains the element 00. It is not empty; it has cardinality 1.

Answer

(a)  ∅,(b)  ∅,(c)  not empty(a)\;\emptyset,\quad (b)\;\emptyset,\quad (c)\;\text{not empty}
The empty set ∅\emptyset contains no elements at all. A set containing zero ({0}\{0\}) is not empty — 00 is a perfectly valid element.

About Empty Set

The empty set, denoted ∅\emptyset or {}\{\}, is the unique set that contains no elements at all. It is a subset of every set because the statement 'every element of ∅\emptyset belongs to AA' is vacuously true — there are no elements to contradict it.

Learn more about Empty Set →

More Empty Set Examples

Example 2 medium

Prove that the empty set [formula] is a subset of every set [formula].

Example 3 easy

Decide which are true: (a) [formula], (b) [formula], (c) [formula].

Example 4 medium

Let [formula]. Find: (a) [formula] for any set [formula], (b) [formula] for any set [formula], (c) [

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Related Concepts

SetCardinality