Symbol meaning

⊂ vs ⊆: Subset or Proper Subset?

Distinguish subset notation that may allow equality from notation that explicitly allows equality or denotes a proper subset.

Compare the possible meanings

NotationTypical meaningHow to identify it
A ⊆ BSubset, equality allowedUnambiguous
A ⊂ BSubset or proper subsetConvention varies
A ⊊ BProper subsetEquality excluded explicitly
A ⊈ BNot a subsetCounterexample element exists

Examples

  • {1,2}⊆{1,2} is true.
  • {1}⊊{1,2} is true.
  • A text using ⊂ should define whether equality is allowed.

How to decide

Prefer ⊆ for subset-with-possible-equality and ⊊ for proper subset when writing for mixed audiences.

How to verify the interpretation

Test both containment and equality, then compare with the document's notation convention.

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