Question

Am nevoie de ajutor la acest exercitiu

Answer 1

$"\bot":\quad x\bot y = \max\{x,y\} \\ "\top": \quad x\top y = \min\{x,y\} \\ \\\bullet Distributivitatea\\ operatiei "\bot" in raport cu operatia "\top": \\ \\1) a\ \leq \ b\ \leq \ c \\ a \bot (b\top c) = a\bot\min\{b,c\} = a\bot b = \max\{a,b\} = b \\ a\bot(b\top c) = (a\bot b) \top(a\bot c) =\\ = \max\{a,b\} \top \max\{a,c\} = b \top c = b \\ \\ 2) b~\leq~ a~\leq~c \\ a\bot (b\top c) = a\bot b = a$
$a\bot (b\top c) = (a\bot b) \top (a\bot c) = a\top c = a \\ \\ 3) a\ \leq \ c\ \leq \ b \\ a\bot (b\top c) = a\bot c = c \\ a\bot (b\top c) = (a\bot b) \top (a\bot c) =b \top c = c \\ \\ 4) c\ \leq \ a\ \leq \ b \\ a\bot (b\top c) = a\bot c = a \\ a\bot (b\top c) = (a\bot b) \top (a\bot c) =b\top a = a \\ \\ 5) c\ \leq \ b\ \leq \ a \\ a\bot (b\top c) = a\bot c = a \\ a\bot (b\top c) = (a\bot b) \top (a\bot c) =a\top a = a \\ \\ 6) b\ \leq \ c\ \leq \ a \\ a\bot (b\top c) = a\bot b = a \\ a\bot (b\top c) = (a\bot b) \top (a\bot c) =a\top a = a$

$\Rightarrow Operatia "\bot" este distributiva \\in raport cu operatia "\top" ! \\ \\ \bullet Distributivitatea\\ operatiei "\top" in raport cu operatia "\bot": \\ \\ 1) a\ \leq \ b\ \leq \ c \\ a\top(b\bot c) = a \top c = a \\ a\top (b\bot c) = (a\top b)\bot (a\top c) = a\bot a = a \\ \\ 2) b\ \leq \ a\ \leq \ c \\ a\top (b\bot c) = a \top c = a \\ a\top (b\bot c) = (a\top b)\bot (a\top c) =b\bot a = a \\ \\ 3) a\ \leq \ c\ \leq \ b \\ a\top (b\bot c) = a\top b = a \\ a\top (b\bot c) = (a\top b)\bot (a\top c) =a\bot a = a$

$4) c\ \leq \ a\ \leq \ b \\ a\top (b\bot c) = a\top b = a \\ a\top (b\bot c) = (a\top b)\bot (a\top c) =a\bot c = a \\ \\ 5) c\ \leq \ b\ \leq \ a \\ a\top(b\bot c) = a\top b = b \\ a\top (b\bot c) = (a\top b)\bot (a\top c) =b\bot c = b \\ \\ 6) b\ \leq \ c\ \leq \ a \\ a\top(b\bot c) = a \top c = c \\ a\top (b\bot c) = (a\top b)\bot (a\top c) = b\bot c = c \\ \\ \Rightarrow Operatia "\top" este distributiva\\ in raport cu operatia "\bot" !$

Daca acele doua relatii de la cel putin un caz, dadeau rezultate diferite, nu am fi avut distributivitate.

Noi intotdeauna am avut a a sau b b sau c c.