Monotonic Function - Boolean Functions

Boolean Functions

In Boolean algebra, a monotonic function is one such that for all a_i and b_i in {0,1}, if a₁ ≤ b₁, a₂ ≤ b₂, ..., a_n ≤ b_n, then f(a₁, ..., a_n) ≤ f(b₁, ..., b_n). In other words, a Boolean function is monotonic if, for every combination of inputs, switching one of the inputs from false to true can only cause the output to switch from false to true and not from true to false. Graphically, this means that a Boolean function is monotonic when in its Hasse diagram (dual of its Venn diagram), there is no 1 (red vertex) connected to a higher 0 (white vertex).

The monotonic Boolean functions are precisely those that can be defined by an expression combining the inputs (which may appear more than once) using only the operators and and or (in particular not is forbidden). For instance "at least two of a,b,c hold" is a monotonic function of a,b,c, since it can be written for instance as ((a and b) or (a and c) or (b and c)).

The number of such functions on n variables is known as the Dedekind number of n.