To simplify our notation, let $\max \set {x, y}$ be (temporarily) denoted $x \overline \wedge y$, and let $\min \set {x, y}$ be (temporarily) denoted $x \underline \vee y$. Note that, once we have proved: :$x \overline \wedge \paren {y \underline \vee z} = \paren {x \overline \wedge y} \underline \vee \paren {x \overline \wedge z}$ :$x \underline \vee \paren {y \overline \wedge z} = \paren {x \underline \vee y} \overline \wedge \paren {x  \underline \vee z}$ then the other results follow immediately by the fact that Min and Max are commutative. There are the following cases to consider: :$(1): \quad x \le y \le z$ :$(2): \quad x \le z \le y$ :$(3): \quad y \le x \le z$ :$(4): \quad y \le z \le x$ :$(5): \quad z \le x \le y$ :$(6): \quad z \le y \le x$ $(1): \quad $ Let $x \le y \le z$. Then: {{begin-eqn}} {{eqn | l = x \overline \wedge \paren {y \underline \vee z}       | r = x \overline \wedge y = y       | c =  }} {{eqn | l = \paren {x \overline \wedge y} \underline \vee \paren {x \overline \wedge z}       | r = y \underline \vee z = y       | c =  }} {{eqn | o =        | c =  }} {{eqn | l = x \underline \vee \paren {y \overline \wedge z}       | r = x \underline \vee z = x       | c =  }} {{eqn | l = \paren {x \underline \vee y} \overline \wedge \paren {x \underline \vee z}       | r = x \overline \wedge x = x       | c =  }} {{end-eqn}} $(2): \quad $ Let $x \le z \le y$. Then: {{begin-eqn}} {{eqn | l = x \overline \wedge \paren {y \underline \vee z}       | r = x \overline \wedge z = z       | c =  }} {{eqn | l = \paren {x \overline \wedge y} \underline \vee \paren {x \overline \wedge z}       | r = y \underline \vee z = z       | c =  }} {{eqn | o =        | c =  }} {{eqn | l = x \underline \vee \paren {y \overline \wedge z}       | r = x \underline \vee y = x       | c =  }} {{eqn | l = \paren {x \underline \vee y} \overline \wedge \paren {x \underline \vee z}       | r = x \overline \wedge x = x       | c =  }} {{end-eqn}} $(3): \quad $ Let $y \le x \le z$. Then: {{begin-eqn}} {{eqn | l = x \overline \wedge \paren {y \underline \vee z}       | r = x \overline \wedge y = x       | c =  }} {{eqn | l = \paren {x \overline \wedge y} \underline \vee \paren {x \overline \wedge z}       | r = x \underline \vee z = x       | c =  }} {{eqn | o =        | c =  }} {{eqn | l = x \underline \vee \paren {y \overline \wedge z}       | r = x \underline \vee z = x       | c =  }} {{eqn | l = \paren {x \underline \vee y} \overline \wedge \paren {x \underline \vee z}       | r = y \overline \wedge x = x       | c =  }} {{end-eqn}} $(4): \quad $ Let $y \le z \le x$. Then: {{begin-eqn}} {{eqn | l = x \overline \wedge \paren {y \underline \vee z}       | r = x \overline \wedge y = x       | c =  }} {{eqn | l = \paren {x \overline \wedge y} \underline \vee \paren {x \overline \wedge z}       | r = x \underline \vee x = x       | c =  }} {{eqn | o =        | c =  }} {{eqn | l = x \underline \vee \paren {y \overline \wedge z}       | r = x \underline \vee z = z       | c =  }} {{eqn | l = \paren {x \underline \vee y} \overline \wedge \paren {x \underline \vee z}       | r = y \overline \wedge z = z       | c =  }} {{end-eqn}} $(5): \quad $ Let $z \le x \le y$. Then: {{begin-eqn}} {{eqn | l = x \overline \wedge \paren {y \underline \vee z}       | r = x \overline \wedge z = x       | c =  }} {{eqn | l = \paren {x \overline \wedge y} \underline \vee \paren {x \overline \wedge z}       | r = y \underline \vee x = x       | c =  }} {{eqn | o =        | c =  }} {{eqn | l = x \underline \vee \paren {y \overline \wedge z}       | r = x \underline \vee y = x       | c =  }} {{eqn | l = \paren {x \underline \vee y} \overline \wedge \paren {x \underline \vee z}       | r = x \overline \wedge z = x       | c =  }} {{end-eqn}} $(6): \quad $ Let $z \le y \le x$. Then: {{begin-eqn}} {{eqn | l = x \overline \wedge \paren {y \underline \vee z}       | r = x \overline \wedge z = x       | c =  }} {{eqn | l = \paren {x \overline \wedge y} \underline \vee \paren {x \overline \wedge z}       | r = x \underline \vee x = x       | c =  }} {{eqn | o =        | c =  }} {{eqn | l = x \underline \vee \paren {y \overline \wedge z}       | r = x \underline \vee y = y       | c =  }} {{eqn | l = \paren {x \underline \vee y} \overline \wedge \paren {x \underline \vee z}       | r = y \overline \wedge z = y       | c =  }} {{end-eqn}} Thus in all cases it can be seen that the result holds. {{qed}} Category:Max and Min Operations Category:Distributive Operations 8aablkhh5eun94ro113f3tfjjhqdm1t