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20 votes
20 votes

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

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6 Answers

Best answer
21 votes
21 votes
  • $lub(a,e) = lub(a,b) = lub(a,c) = lub(a,d) = I$ (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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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

 

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.

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 $\overline{a}$

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