Unate function

A unate function is a type of

boolean function In mathematics, a Boolean function is a function whose arguments and result assume values from a two-element set (usually , or ). Alternative names are switching function, used especially in older computer science literature, and truth function ( ...

which has

monotonic In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order ...

properties. They have been studied extensively in

switching theory Switching circuit theory is the mathematical study of the properties of networks of idealized switches. Such networks may be strictly combinational logic, in which their output state is only a function of the present state of their inputs; or may ...

. A function

f(x_1,x_2,\ldots,x_n)

is said to be positive unate in

x_i

if for all possible values of

x_j

j\neq i

f(x_1,x_2,\ldots,x_,1,x_,\ldots,x_n) \ge f(x_1,x_2,\ldots,x_,0,x_,\ldots,x_n).\,

Likewise, it is negative unate in

x_i

if :

f(x_1,x_2,\ldots,x_,0,x_,\ldots,x_n) \ge f(x_1,x_2,\ldots,x_,1,x_,\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 (note that some

x_i

may be positive unate and some negative unate to satisfy the definition of unate function). A function is binate if it is not unate (i.e., is neither positive unate nor negative unate in at least one of its variables). For example, the

logical disjunction In logic, disjunction is a logical connective typically notated as \lor and read aloud as "or". For instance, the English language sentence "it is raining or it is snowing" can be represented in logic using the disjunctive formula R \lor S ...

function ''or'' with boolean values used for true (1) and false (0) is positive unate. Conversely,

Exclusive or Exclusive or or exclusive disjunction is a logical operation that is true if and only if its arguments differ (one is true, the other is false). It is symbolized by the prefix operator J and by the infix operators XOR ( or ), EOR, EXOR, , ...

is non-unate, because the transition from 0 to 1 on input x0 is both positive unate and negative unate, depending on the input value on x1. 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) Syntax (logic) {{Compu-lang-stub