TeXipedia

cong

Denotes geometric congruence or isomorphism between mathematical objects, indicating they have the same shape and size.

Overview

Essential in geometry, abstract algebra, and topology for expressing equivalence relationships between objects that preserve specific properties.

  • Commonly used in geometric proofs to show that two figures are identical in shape and size but may have different positions or orientations.
  • Appears frequently in group theory and ring theory to indicate isomorphic structures.
  • Useful in mathematical education when discussing similarity transformations and rigid motions.
  • Distinguished from equality (=) as it specifically denotes preservation of geometric or structural properties rather than exact sameness.

Examples

Showing that two triangles are congruent in geometry.

△ABC≅△DEF\triangle ABC \cong \triangle DEF 
\triangle ABC \cong \triangle DEF

Expressing congruence in modular arithmetic.

15≅3(mod12)15 \cong 3 \pmod{12} 
15 \cong 3 \pmod{12}

Demonstrating isomorphic groups in abstract algebra.

G/H≅KG/H \cong K 
G/H \cong K