Signs Used in Set Theory
April 14, 2025•613 words
Signs Used in Set Theory
1. Membership and Non-Membership
| Sign | Name | Meaning |
|---|---|---|
| ∈ | Element of | x ∈ A: x is an element of A |
| ∉ | Not an element of | x ∉ A: x is not an element of A |
2. Set Inclusion and Equality
| Sign | Name | Meaning |
|---|---|---|
| ⊆ | Subset | A ⊆ B: A is a subset of B |
| ⊂ | Proper subset | A ⊂ B: A is a proper subset of B |
| ⊇ | Superset | A ⊇ B: A is a superset of B |
| ⊃ | Proper superset | A ⊃ B: A properly contains B |
| = | Equality | A = B: A and B are equal |
| ≠ | Inequality | A ≠ B: A and B are not equal |
3. Set Operations
| Sign | Name | Meaning |
|---|---|---|
| ∪ | Union | A ∪ B: all elements in A or B or both |
| ∩ | Intersection | A ∩ B: all elements common to A and B |
| \ | Set difference | A \ B: elements in A but not in B |
| ∆ | Symmetric difference | A ∆ B: elements in A or B, but not both |
| × | Cartesian product | A × B: set of ordered pairs (a, b) where a ∈ A, b ∈ B |
4. Special Sets
| Sign | Name | Meaning |
|---|---|---|
| ∅ | Empty set | The set with no elements |
| ℕ | Natural numbers | {0, 1, 2, 3, ...} |
| ℤ | Integers | {..., -2, -1, 0, 1, 2, ...} |
| ℚ | Rational numbers | {p/q |
| ℝ | Real numbers | The set of real numbers |
| ℂ | Complex numbers | The set of complex numbers |
| 𝒫(A) | Power set | The set of all subsets of A |
5. Logical Symbols Used in Set Theory
| Sign | Name | Meaning |
|---|---|---|
| ∃ | Existential quantifier | "There exists" |
| ∀ | Universal quantifier | "For all" |
| ⇒ | Implication | "If ..., then ..." |
| ⇔ | Biconditional | "If and only if" |
| ¬ | Negation | "Not" |
| ∧ | Conjunction | "And" |
| ∨ | Disjunction | "Or" |
| ⊢ | Provable from | Used in derivations |
6. Brackets and Delimiters
| Sign | Name | Usage |
|---|---|---|
| { } | Set brackets | Used to define sets, e.g., {1, 2, 3} |
| ( ) | Parentheses | Used for grouping or ordered pairs |
| [ ] | Interval notation | [a, b] means closed interval from a to b |
| ⟨ ⟩ | Angle brackets | Used for tuples or sequences |
7. Functions and Maps
| Sign | Name | Meaning |
|---|---|---|
| → | Maps to | f: A → B: f maps elements of A to elements of B |
| ↦ | Mapping | x ↦ f(x): defines mapping rule |
8. Optional: Category Theory Notation
| Sign | Name | Usage |
|---|---|---|
| ∅ → A | Initial morphism | Unique map from empty set |
| A × B | Product | Categorical product of A and B |