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Implication and equivalence

An implication says that if the first statement is true, then the second is also true. It says nothing about the way back. Equivalence holds both ways, and then you may replace one statement by the other in an equation. One counterexample is enough to show that an implication is false. $x^2 = 9$ follows from $x = 3$, but the converse is false because $-3$ also gives $9$.

Symbols

Example

$2x + 1 = 7 \Leftrightarrow x = 3$, since $(7 - 1)/2 = 3$. $x = 3 \Rightarrow x^2 = 9$, but $x = -3$ also gives $9$, so the arrow cannot be reversed.

← A circle in parametric form · Direct proof →

Part of Mathematics R1: Logic and proof.