Empty Set Math Example 2

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Example 2

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

Solution

  1. 1
    By definition, ∅⊆A\emptyset \subseteq A means: for every xx, if x∈∅x \in \emptyset then x∈Ax \in A.
  2. 2
    Since ∅\emptyset has no elements, the condition 'x∈∅x \in \emptyset' is always false.
  3. 3
    A conditional with a false hypothesis is vacuously true. Therefore ∅⊆A\emptyset \subseteq A for any set AA.

Answer

∅⊆A for every set A\emptyset \subseteq A \text{ for every set } A
The subset condition ∅⊆A\emptyset \subseteq A is vacuously true: there are no elements in ∅\emptyset that could fail to be in AA. This is a fundamental property of the empty set.

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 1 easy

Determine whether each set is empty: (a) [formula], (b) [formula], (c) [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