TeXipedia

implies

Represents logical implication, indicating that one statement leads to or necessitates another in mathematical proofs and logical arguments.

Overview

Essential in mathematical logic, proof writing, and formal reasoning where conditional relationships need to be expressed clearly.

  • Commonly used in discrete mathematics and formal logic to show that one proposition leads to another
  • Appears frequently in theorem statements and their proofs
  • Important in computer science for expressing logical dependencies and program specifications
  • Often paired with other logical operators to construct complex logical statements and mathematical arguments

Examples

Logical implication in a conditional statement

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

Mathematical proof showing implication between sets

A⊆B  ⟹  ∣A∣≤∣B∣A \subseteq B \implies |A| \leq |B| 
A \subseteq B \implies |A| \leq |B|

Logical statement in number theory

n is prime  ⟹  n is odd or n=2n \text{ is prime} \implies n \text{ is odd or } n = 2 
n \text{ is prime} \implies n \text{ is odd or } n = 2