Group theory · Definition

Subset

sets

Before you start you need to know

A subset expresses that one set is contained in another: everything in the first set also belongs to the second.

Definition

A set A is a subset of B when every element of A also belongs to B.

\[A \subseteq B \Longleftrightarrow \forall x\,(x\in A \Rightarrow x\in B)\]

Every set is a subset of itself, and the empty set is a subset of every set. The relation is transitive: if \( A \subseteq B \) and \( B \subseteq C \), then \( A \subseteq C \).

With \( A = \{1,2\} \) and \( B = \{1,2,3\} \) we have \( A \subseteq B \), because 1 and 2 belong to B. By contrast \( B \not\subseteq A \), because 3 does not belong to A.

A common mistake is confusing ⊆ with ∈: \( \{1\} \subseteq A \) means that 1 is in A, while \( \{1\} \in A \) means that the set \{1\} is an element of A.

Think about it

Why does ∅ ⊆ A hold for every set A?

Because there is no element of ∅ that could violate the condition; the implication is vacuously true.

Exercises 10

  1. If A = {1,2} and B = {1,2,3}, which relation is true?
  2. A ⊆ B means that...
  3. Which statement is false?
  4. For any set A, it always holds that...
  5. The empty set is a subset of...
  6. If A ⊆ B and B ⊆ C, what can be concluded?
  7. A ⊂ B usually indicates that...
  8. How many subsets does {a,b} have?
  9. Which statement is always true for any set A?
  10. If 1 ∈ A, which statement does not necessarily follow?

References

  • I. N. Herstein, Topics in Algebra, 2.ª edición, John Wiley & Sons, 1975.