TeXipedia

impliedby

Represents logical implication in reverse direction, indicating that the left statement follows from the right statement.

Overview

Essential in mathematical proofs, logical reasoning, and formal mathematics where reverse implications need to be shown explicitly.

  • Common in discrete mathematics and formal logic to show necessary conditions.
  • Often paired with \implies to demonstrate logical equivalence.
  • Particularly useful in theorem proving and mathematical analysis where step-by-step reasoning needs to be clearly documented.
  • Frequently appears in advanced mathematics textbooks and academic papers where precise logical relationships must be expressed.

Examples

Showing the logical implication that a mathematical statement follows from another condition.

x>0  ⟸  x2>0x > 0 \impliedby x^2 > 0 
x > 0 \impliedby x^2 > 0

Expressing a necessary condition in set theory.

A⊆B  ⟸  ∀x(x∈A→x∈B)A \subseteq B \impliedby \forall x(x \in A \rightarrow x \in B) 
A \subseteq B \impliedby \forall x(x \in A \rightarrow x \in B)

Demonstrating reverse implication in a proof by contradiction.

n is prime  ⟸  n has exactly two factorsn \text{ is prime} \impliedby n \text{ has exactly two factors} 
n \text{ is prime} \impliedby n \text{ has exactly two factors}