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Longrightarrow

Represents a logical implication or material conditional in mathematical proofs and logical statements.

Overview

Serves as a formal symbol in mathematical logic, set theory, and proof writing to indicate that one statement necessarily leads to another.

  • Commonly used in theorem proofs to show logical consequences
  • Appears frequently in formal mathematical writing to denote stronger implications than the standard arrow
  • Essential in axiomatic systems and formal logic where precise relationship between statements must be shown
  • Often used in conjunction with other logical symbols in complex mathematical arguments

Examples

Logical implication in a mathematical statement showing that one condition leads to another.

x>0⟹x2>0x > 0 \Longrightarrow x^2 > 0 
x > 0 \Longrightarrow x^2 > 0

Showing the steps of a mathematical derivation with detailed reasoning.

(x+y)2=x2+2xy+y2⟹x2+y2≥−2xy(x+y)^2 = x^2 + 2xy + y^2 \Longrightarrow x^2 + y^2 \geq -2xy 
(x+y)^2 = x^2 + 2xy + y^2 \Longrightarrow x^2 + y^2 \geq -2xy

Expressing a logical conclusion in set theory.

A⊆B and B⊆C⟹A⊆CA \subseteq B \text{ and } B \subseteq C \Longrightarrow A \subseteq C 
A \subseteq B \text{ and } B \subseteq C \Longrightarrow A \subseteq C