2 votes 2 votes 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 ? $[N, \leq ]$ is distributive but not complemented lattice $[N, \leq ]$ is not a lattice $[N, \leq ]$ is not Boolean lattice Element $0$ doesn't have complement Set Theory & Algebra goclasses_wq10 goclasses set-theory&algebra partial-order lattice 1-mark + – GO Classes asked May 12, 2022 GO Classes 557 views answer comment Share Follow See 1 comment See all 1 1 comment reply Kushal06 commented May 14, 2022 reply Follow Share POR: Partial ordered relationGLB: Greatest lower boundLUB: least upper boundLet’s analyze the optionsB) FalseIt is a Lattice ,since for every pair of elements GLB and LUB exist.A) TrueSince 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 ,It should be bounded i.e. It should have least element and greatest element.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) TrueFor 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) TrueFor 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 complementFinally we are asked about False statement . Hence Option B is correct option. 2 votes 2 votes Please log in or register to add a comment.
1 votes 1 votes $[N, \leq ]$ is a total order, so it is a distributive lattice, but not complemented. GO Classes answered May 12, 2022 GO Classes comment Share Follow See all 0 reply Please log in or register to add a comment.
