Monotone Boolean Function

A Monotone Boolean Function is a boolean function that is a monotone function such that for all a_i and b_i in {0,1}, if a₁ ≤ b₁, a₂ ≤ b₂, ..., a_n ≤ b_n (i.e. the Cartesian product {0, 1}ⁿ is ordered coordinatewise), then f(a₁, ..., a_n) ≤ f(b₁, ..., b_n).

Example(s):
- …
Counter-Example(s):
- Non-Monotone Boolean Function.
- Monotone Continuous Function.
See: Order Theory, Voting Systems, Mathematical Order Structures, Order Relation.

References

(Wikipedia, 2017) ⇒ https://en.wikipedia.org/wiki/Monotonic_function#Boolean_functions Retrieved:2017-8-14.
- 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 (i.e. the Cartesian product {0, 1}ⁿ is ordered coordinatewise), 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 connected to a higher 0.
  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.