Editing boolean algebra/majority function

'''Majority function''' (sometimes '''quorum function''') is a [[threshold function]] that produces a 1 if and only if the majority of the inputs are 1. Otherwise, the output is 0. This function is only defined for three or more odd inputs. The majority function can be found in various applications such as [[adder]]s, [[subtractor]]s, [[cryptographic hash function|hash functions]], and [[Muller C-element]].

:<math>
\text{MAJ}(x_1,x_2,...,x_n) =
\begin{cases}
1, & \text{if} \sum^{n}_{i=1}x_i \ge \frac{n}{2} \\[2ex]
0, & \text{otherwise.}
\end{cases}
</math>

A 3-input majority function can be implemented using the following Boolean function <math>\text{MAJ}(x,y,z) = (x \land y) \oplus (x \land z) \oplus (y \land z)</math>.
@@ Line 1: / Line 1: @@
-{{ba title|Majority Function (Maj)}}
-{| class="wikitable" style="float: right; text-align: center;"
-! colspan="4" | 3-input Majority
-|-
-! colspan="3" | Inputs !! Output
-|-
-! X !! Y !! Z !! Q
-|-
-| 0 || 0 || 0 || 0
-|-
-| 0 || 0 || 1 || 0
-|-
-| 0 || 1 || 0 || 0
-|-
-| 0 || 1 || 1 || 1
-|-
-| 1 || 0 || 0 || 0
-|-
-| 1 || 0 || 1 || 1
-|-
-| 1 || 1 || 0 || 1
-|-
-| 1 || 1 || 1 || 1
-|}
 '''Majority function''' (sometimes '''quorum function''') is a [[threshold function]] that produces a 1 if and only if the majority of the inputs are 1. Otherwise, the output is 0. This function is only defined for three or more odd inputs. The majority function can be found in various applications such as [[adder]]s, [[subtractor]]s, [[cryptographic hash function|hash functions]], and [[Muller C-element]].
@@ Line 33: / Line 9: @@
 </math>
-== 3-input Majority Function==
+A 3-input majority function can be implemented using the following Boolean function <math>\text{MAJ}(x,y,z) = (x \land y) \oplus (x \land z) \oplus (y \land z)</math>.
-A 3-input majority function is defined as:
-:<math>f:\mathbb{B}^3 \to \mathbb{B}, \text{ is } \text{MAJ3}(x,y,z) = (x \land y) \lor (x \land z) \lor (y \land z)</math>
-In the field of [[cryptography]], the XOR version is also common.
-:<math>\text{MAJ3}(x,y,z) = (x \land y) \oplus (x \land z) \oplus (y \land z)</math>
-=== properties ===
-The majority function is a [[unate function]], symmetric, monotone increasing, and self-dual. It therefore, with the addition of just an [[inverter]] it can satisfy all the conditions needed to be [[functionally complete]] (i.e. '''{'''[[not gate|NOT]], [[MAJ]]'''}''' is a complete set). Being self-dual means that <math>\text{MAJ}(\bar a, \bar b, \bar c) = \overline{\text{MAJ}(a, b, c)}</math> which could yields various hardware implementation optimization - such as floating the [[inversion bubble|inversion point]] to a more desired location.
-=== median algebra ===
-{{main|median algebra}}
-By treating the majority function [[conjunction]] and [[disjunction]] as min and max functions respectively, then it's easy to see how the majority function can serve as the median - i.e. <math>\text{MAJ3}(x,y,z) = y \text{ if } x \le y \le z</math>. '''Median algebra''' is a generalized idea based on the majority function, algebra with axioms on a set of [[truth value]]s.
-== Majority gate ==
-{{main|majority gate}}
-The '''[[majority gate]]''' is a [[logic gate]] that implements the majority function as a circuit.
-== See also ==
-* [[threshold function]]