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# DM - Grand Test -Q6

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+1 vote

A lattice is called a if it is both and. Which of the following statement is false in case of a

(A) In a every element will have a unique complement.

(B) The poset is a boolean algebra , where is the set of all divisors of and represents the relation.

(C) Let A= and represents the power set of , then the poset where represents the subset relation is a boolean algebra.

(D) Every total ordered relation is a boolean algebra.

reshown Aug 31, 2020

A TOS on n elements is BA iff n=2. So, Option D is false.

In a distributive lattice each element has atmost one complement (that is either no complements or 1 complement.

In a lattice if upper bound and lower bound exists then it is called a bounded lattice. Let L be a bounded lattice, if each element of L has complement in L , then L is called a complemented lattice. In a complemented lattice each element has atleast one complement. A lattice is boolean algebra if it is both distributive and complemented. So in a boolean algebra each element will have exactly one complement.

$\textbf{Option B : True} \newline D_{110} = \{1 , 2, 5 , 10, 11, 22, 55, 110\}$

The hasse diagram is identical to hasse diagram for the poset $[P(A) ; \subseteq ]$ So its a boolean algebra. Complement of 10 is 11, Complement of 110 is 1, Complement of 22 is 5, Complement of 55 is 2, And vice - versa.

$\textbf{Option C : True} \newline \newline$

$\text{ The Hasse diagram for the poset} [P(A) ; \subseteq] is \,\, given \,\, below.$

Every total ordered relation is distributive but a total order relation which contain more than 2 elements cant be complemented since in a total ordered chain elements are directly related , so we wont get complements for elements other than Upper bound and lower bound.

Option D is the correct answer

answered Sep 9, 2020 by (226,240 points)
amazing explanation, with crystal clear concepts.