binaryvalued

Binary-valued, typically written as binary-valued or binary valued, refers to quantities, variables, or functions that can take exactly two possible values. The two values are usually denoted as 0 and 1, false and true, or off and on. This dichotomy is foundational in logic, computer science, and many applied disciplines.

In mathematics and logic, a binary-valued function maps its inputs to the set {0,1}. Such functions are often called indicator or characteristic functions, signaling whether a given element belongs to a particular subset. In propositional logic, a valuation assigns to each proposition a truth value of 0 or 1, forming the basis of Boolean reasoning and Boolean algebra, where logical operations correspond to arithmetic on binary values.

In statistics and data analysis, a binary-valued variable (also called a dummy variable or indicator variable) encodes a dichotomous category, such as male/female or success/failure. These variables are common in regression models and machine learning, where they enable the inclusion of qualitative information in quantitative analyses.

In computing and digital electronics, binary-valued logic is implemented in circuits using two voltage levels or states. Boolean logic operations—AND, OR, NOT—manipulate binary-valued signals, forming the core of modern digital processors and data representation.

Limitations of the concept arise when real-world phenomena cannot be accurately captured by two states, necessitating multi-valued or continuous representations. Nevertheless, binary-valued frameworks remain essential for clear, efficient modeling and computation.