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forall

Represents the universal quantifier in mathematical logic, indicating that a statement holds true for all elements in a given set.

Overview

Essential in formal mathematics and logic for expressing universal statements and properties that apply across entire domains.

  • Commonly used in set theory to define properties that hold for all members of a set
  • Appears frequently in mathematical proofs and formal definitions
  • Important in computer science for specifying program invariants and formal verification
  • Often paired with implications to express conditional statements that apply universally

Examples

Universal quantification in a logical statement about real numbers.

∀x∈R,x2≥0\forall x \in \mathbb{R}, x^2 \geq 0 
\forall x \in \mathbb{R}, x^2 \geq 0

Mathematical proposition about even numbers.

∀n∈N,2n is even\forall n \in \mathbb{N}, 2n \text{ is even} 
\forall n \in \mathbb{N}, 2n \text{ is even}

Set theory statement about subset relationships.

∀A⊆X,A∩∅=∅\forall A \subseteq X, A \cap \emptyset = \emptyset 
\forall A \subseteq X, A \cap \emptyset = \emptyset