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20 votes
20 votes

The complement(s) of the element $'a'$ in the lattice shown in below figure is (are) ____

in Set Theory & Algebra recategorized by
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3 Comments

c,b,e,d
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is this an example of a complemented lattice??
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amitarp818 yes it is a complemented lattice.

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6 Answers

21 votes
21 votes
Best answer
  • $lub(a,e) = lub(a,b) = lub(a,c) = lub(a,d) = I$ (Upper Bound of Lattice)
  • $glb(a,e) = glb(a,b) = glb(a,c) = glb(a,d) = O$ (Lower Bound of Lattice)


So, $e, b, c, d$ all are complement of $a.$

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4 Comments

How b is possible to complement of a..I don't get it
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@mr robot

lub(a,b)=I (Upper Bound of Lattice)

glb(a,b)=O (lower Bound of Lattice)
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The given lattice is complemented lattice but not distributive;

  • $\bar a=(b,c,d,e)$
  • $\bar b=(a,d,e)$
  • $\bar c=(a,d,e)$
  • $\bar d=(a,b,c,e)$
  • $\bar e=(a,b,c,d)$

Please verify it.

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It is true @Son_of_Thakur_Bhanu_Pratap_Singh because in a complemented lattice number of complements of an element should be less than or equal to 1.
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5 votes
5 votes

Imagine you are sitting at a and your friend is sitting at d, imagine in your and your friends world, there is gravity on both the ends the top end and the bottom end, now you and your friend share a special property that if you decide to jump in upward direction or fall in lower direction, you will shake hands with each other

If you are able to say that you have find such a friend, then that friend is your complement

Now try to relate this story to the definition of GLB and LUB and if you and your friend both GLB and LUB coincide, then my friend 

Congratulations, you have found your complement

 

1 comment

Good one.
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4 votes
4 votes

An element $b\in L$ is complement of $a\in L$ if 

$lub\{a,b\}=a\vee b=a\ join\ b=1(greatest)$

$glb\{a,b\}=a\wedge b=a\ meet\ b=0(least)$


$a\vee d=I(greatest)$

$a\wedge d=O(least)$


$a\vee b=I(greatest)$

$a\wedge b=O(least)$


$a\vee c=I(greatest)$

$a\wedge c=O(least)$


$a\vee e=I(greatest)$

$a\wedge e=O(least)$

 

Correct answer : d,b,c,e.

3 Comments

@Kushagra गुप्ता

join is upper bound or least upper bound?

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thanks..
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2 votes
2 votes

LUB of $a$ and $\overline{a}$ must be $I$

GLB of $a$ and $\overline{a}$ must be $O$

where $I$ is the upper bound, and where $O$ is the lower bound

 

All of d,b,c,e qualify to be $\overline{a}$

1 comment

edited by
Can $I$ and $O$ also be complements? What are the LUBs and GLBs of $\{a, I\}$ and $\{a, O\}$?

Edit: Okay cleared self-doubt: GLB of first one is $a$ itself, not the LB of the lattice, so they are not a complement pair. Similarly for the LUB of the second.
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