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Let $[N, \leq ]$ is a partial order relation defined on natural numbers, where “$\leq$” is the “less than equal to” relation defined on $N = \{ 0,1,2,3,\dots \}.$ Which of the following statements is false ?
  1. $[N, \leq ]$ is distributive but not complemented lattice
  2. $[N, \leq ]$ is not a lattice
  3. $[N, \leq ]$ is not Boolean lattice
  4. Element $0$ doesn't have complement
in Set Theory & Algebra
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POR: Partial ordered relation

GLB: Greatest lower bound

LUB: least upper bound

Let’s analyze the options

B) False

It is a Lattice ,since for every pair of elements GLB and LUB exist.

A) True

Since neither pentagon nor kite structure will never be possible in sublattice of given lattice , as it is total order lattice. Thus given Lattice is Distributive lattice .

For a lattice to be called complemented lattice it should satisfy two main properties ,

  1. It should be bounded i.e.  It should have least element and greatest element.

  2. GLB and LUB performed on any pair of element of the given set should always give respective least element and the greatest element , simultaneously.

Since given relation defined on a set is Total ordered  relation. In addition , it is not bounded. We know that in total ordered reation (given in Question) greatest element doesn’t seems to be fixed. Thus it is not complemented Lattice.

C) True

For a Lattice to be called boolean lattice it should have properties such as bounded,complemented and distributive.

Given POR is not bounded ,hence no question of boolean lattice.

D) True

For an element to have complement , its GLB and LUB with the paired element should fetch least element and greatest Element respectively .

Greatest element is not fixed i.e. notbounded/infinite .

Thus Element O doesn’t have complement


Finally we are asked about False statement . Hence Option  B is correct option.


 

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1 Answer

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$[N, \leq ]$ is a total order, so it is a distributive lattice, but not complemented.