Symbols

  • Quantifiers:
    • Universal quantifier: ∀, /\: for every, for all
    • Existential quantifier: Ǝ, \/: there exists a … such that, there is some … such that
    • Example: (∀ y)(Ǝ x)[x > y] is symbolic for “for all y there is some x with x > y”.
  • Logical connectives:
    • Negation: ¬, ~ : not
    • Conjunction: ∧, &: and
    • Disjunction: ∨, v: or
    • Implication: →, –>: implies, if … , then … .
    • Biconditional: ↔, : if and only if
    • Logical equivalence: ≡
  • Set theoretic symbols
    • Relations
      • Membership relation:
        • ∈: is an element of
        • ∉: is not an element of
        • ∋: has as a member the element
      • Subset relation: ⊆, ⊂
      • Superset relation: ⊇, ⊃
      • Not a subset of relation: ⊄, ⊈
      • Order relation: ≤, ≥
    • Operations:
      • Union: ∪, ⋃
      • Intersection: ∩, ⋂
      • Set difference: −, \
      • Cartesian product: ×
      • Power set (Weierstrass p): ℘ or ℙ
    • Special sets
      • The emptyset: ∅, {}
      • Set of real numbers: ℜ or ℝ
      • Set of rational numbers: ℚ = { m/n | m, n ∈ ℤ and n>0 }
      • Set of natural numbers: ℕ = { 0, 1, 2, … }
      • Set of positive integers: ℤ+ = { 1, 2, 3, … }
      • Set of integers: ℤ = { … , -3, -2, -1, 0, 1, 2, 3, … }
      • Interval notation: (-∞,0], [0,1), (1,+∞).
    • Set bracket notation: { x | property P(x) } is symbolic for “the set of all x such that property P(x) holds”.
  • Other mathematical symbols
    • Summation: ∑
    • Product: ∏
    • Cardinals: ℵ0, ℵ1, . . . , ℵn,
      ℵω
    • Families of sets: ℬ, ℱ, ℋ, ℒ, ℳ, ℛ, \(\mathcal{A}, \mathcal{B}, \mathcal{C}, \mathcal{D}, \mathcal{E}, \mathcal{F}, \mathcal{G}, \mathcal{H}, \mathcal{I}, \mathcal{J}, \mathcal{K}, \mathcal{L}, \mathcal{P}, \mathcal{Q}, \mathcal{R}, \mathcal{S},\mathcal{T}\)