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DM - Grand Test -Q6


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+1 vote

A lattice L is called a boolean\text{ } algebra if it is both distributive \text{ }lattice andcomplemented \text{ }lattice. Which of the following statement is false in case of a $boolean$ \text{ }$algebra$ ?

(A) In a boolean \text{ }algebra every element will have a unique complement.

(B) The poset [D_{110} ; /] is a boolean algebra , where D_{110} is the set of all divisors of 110 and '$/$' represents the divides relation.

(C) Let A=\{a , b, c\} and P(A) represents the power set of A , then the poset [P(A); \subseteq] where '\subseteq' represents the subset relation is a boolean algebra.

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

asked Aug 30, 2020 by gbeditor-2 (72,840 points)
reshown Aug 31, 2020 by gbeditor-2

1 Answer

0 votes
 
Best answer

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\}

image

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.

image

 

 

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 deepak-gatebook (226,240 points)
amazing explanation, with crystal clear concepts.
Your answer is enough, :)
Answer:

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