Unate function

A unate function is a type of boolean function which has monotonic properties. They have been studied extensively in switching theory.

A function $f(x_{1},x_{2},\ldots ,x_{n})$ is said to be positive unate in $x_{i}$ if

$f(x_{1},x_{2},\ldots ,x_{i-1},1,x_{i+1},\ldots ;x_{n})>=f{\bar {(}}x_{1},x_{2},\ldots ,x_{i-1},0,x_{i+1},\ldots ,x_{n}).\,$

Likewise, it is negative unate in $x_{i}$ if

$f{\bar {(}}x_{1},x_{2},\ldots ,x_{i-1},0,x_{i+1},\ldots ;x_{n})>=f(x_{1},x_{2},\ldots ,x_{i-1},1,x_{i+1},\ldots ,x_{n}).\,$

If for every $x_{i}$ f is either positive or negative unate in the variable $x_{i}$ then it is said to be unate. A function is binate if it is not unate.

For example the Logical disjunction function or with boolean values are used for true (1) and false (0) is positive unate.