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In the lattice defined by the Hasse diagram given in following figure, how many complements does the element ‘$e$’ have?

  1. $2$
  2. $3$
  3. $0$
  4. $1$
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Answer: B

Complement of an element $a$ is $a'$ if:

  • $a ∧ a' = 0$ (lowest vertex in the Hasse diagram)
  • $a ∨ a' = 1$ (highest vertex in the Hasse diagram)

$g, c$ and $d$ are the complements of $e.$

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Option B

vertex 'e' have three complement like 'g','c' and 'd'.

When it will take LUB or GLB with g ,c or d get same answer.

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Another easier way of solving is:=

Find out the greatest and least element, Here it is a and f respectively

For complements, check if the LUB and GLB of any 2 elements (here e with all the other elements) coincide with the greatest and least element respectively.

Here, (e,c)(e,g)(e,d) satisfies the above condition i.e. the LUB and GLB coincide with the greatest and least elements for all the 3 pairs.

But for (e,f), The upper bounds are a,b,e, and the LUB is e. But LUB != maximal element, so it is not a complement.

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