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Downarrow

Represents a double vertical downward-pointing arrow, commonly used in mathematical logic and set theory to indicate logical implications or relationships.

Overview

Serves as a distinctive vertical arrow notation in mathematical expressions, particularly valuable in formal logic, proof writing, and theoretical mathematics.

  • Often used to denote logical consequences or downward relationships in formal proofs
  • Appears in set theory to show mappings or hierarchical relationships
  • Frequently employed alongside its upward counterpart (\Uparrow) for bidirectional relationships
  • Distinguished from the single arrow (\downarrow) by its more emphatic, formal appearance

Examples

Showing logical implication with double down arrow in a formal logic expression

p⇓q≡¬p↑¬qp \Downarrow q \equiv \neg p \uparrow \neg q 
p \Downarrow q \equiv \neg p \uparrow \neg q

Representing downward vector components in a physics equation

F⇓=−mgF_{\Downarrow} = -mg 
F_{\Downarrow} = -mg

Displaying a system of equations with a double vertical arrow

x+y=2⇓2x+2y=4\begin{array}{rcl} x + y & = & 2 \\ \Downarrow \\ 2x + 2y & = & 4 \end{array} 
\begin{array}{rcl} x + y & = & 2 \\ \Downarrow \\ 2x + 2y & = & 4 \end{array}