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ncong

Indicates non-congruence between mathematical expressions or geometric figures, serving as the negation of the congruence relation.

Overview

Essential in mathematical proofs and geometric reasoning where demonstrating non-congruence is required.

  • Common in abstract algebra for showing relationships between algebraic structures
  • Used in geometry to express that shapes or figures do not have the same size and shape
  • Appears frequently in number theory when discussing modular arithmetic
  • Helpful in educational contexts when teaching equivalence relationships and their negations

Examples

Showing that two triangles are not congruent based on their side lengths.

△ABC≆△DEF\triangle ABC \ncong \triangle DEF 
\triangle ABC \ncong \triangle DEF

Demonstrating non-congruence of remainders in modular arithmetic.

5≆2(mod3)5 \ncong 2 \pmod{3} 
5 \ncong 2 \pmod{3}

Expressing that two algebraic expressions are not congruent under a specific modulus.

x2+1≆x+1(mod2)x^2 + 1 \ncong x + 1 \pmod{2} 
x^2 + 1 \ncong x + 1 \pmod{2}