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How many among A),B),C),D) are distributive lattices ?

A) [ P(A),$\subseteq$] where A is a set with n elements.

B) [ P(A),$\subset$] where A is a set with n elements.

C)  [ P(A),$\supset$] where A is a set with n elements.

D)  [ P(A),$\supseteq$] where A is a set with n elements.

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please verify

B) and C) are not partial orders itself,since they are not reflexive.So they cant be lattices and so cant be distributive lattices.

A) is a distributive lattices,because 

         In A) LUB is nothing but UNION operation and GLB is nothing but INTERSECTION operation and we know that "union is distributive over intersection and intersection is distributive over union" from set theory. So we can say that LUB is distributive over GLB and vice-versa.

B) is a distributive lattice because

        In B) LUB is nothing but INTERSECTION operation and GLB is nothing but UNION operation.Again from set theory we already know that  "union is distributive over intersection and intersection is distributive over union". 

Also if someone can provide some more proofs for A) and D) to be called as distributive lattices,it will be helpful.

 

.

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