# Unate function

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

${\displaystyle f(x_{1},x_{2},\ldots ,x_{i-1},1,x_{i+1},\ldots ,x_{n})\geq f(x_{1},x_{2},\ldots ,x_{i-1},0,x_{i+1},\ldots ,x_{n}).\,}$

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

${\displaystyle f(x_{1},x_{2},\ldots ,x_{i-1},0,x_{i+1},\ldots ,x_{n})\geq f(x_{1},x_{2},\ldots ,x_{i-1},1,x_{i+1},\ldots ,x_{n}).\,}$

If for every ${\displaystyle x_{i}}$ f is either positive or negative unate in the variable ${\displaystyle x_{i}}$ then it is said to be unate (note that some ${\displaystyle x_{i}}$ may be positive and some negative to satisfy the definition of unate). A function is binate if it is not unate (i.e., is neither positive nor negative in at least one of its variables).

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

NB: positive unateness can also be considered as passing the same slope (no change in the input) and negative unate is passing the opposite slope.... non unate is dependence on more than one input (of same or different slopes)

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