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equalscoloncolon

Represents a definition or declaration relationship, commonly used in mathematical proofs and formal specifications.

Overview

Serves as a specialized equality symbol that explicitly indicates when something is being defined or declared, particularly useful in formal mathematics and computer science contexts.

  • Often employed in mathematical logic and set theory to introduce new definitions
  • Frequently appears in formal specifications and programming language semantics
  • Helps distinguish definitional equality from standard equality relationships
  • Provides clearer visual separation than simple equals signs when establishing definitions

Examples

Defining a function using the equalscoloncolon symbol to indicate a definition.

f(x)=∷x2+2x+1f(x) \equalscoloncolon x^2 + 2x + 1 
f(x) \equalscoloncolon x^2 + 2x + 1

Introducing a new mathematical operator with a formal definition.

dist⁡(x,y)=∷(x2−x1)2+(y2−y1)2\operatorname{dist}(x,y) \equalscoloncolon \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} 
\operatorname{dist}(x,y) \equalscoloncolon \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}

Defining a sequence using explicit notation.

an=∷1n2for n≥1a_n \equalscoloncolon \frac{1}{n^2} \quad \text{for } n \geq 1 
a_n \equalscoloncolon \frac{1}{n^2} \quad \text{for } n \geq 1