0 votes 0 votes Q)which of the following is not a distributive lattice? a) [P(A);$\preceq$ ] where A = { a,b,c,d } b) [ {1,2,3,5,30} ; / ] Set Theory & Algebra discrete-mathematics set-theory&algebra lattice + – Lakshman Bhaiya asked Mar 17, 2018 Lakshman Bhaiya 2.9k views answer comment Share Follow See 1 comment See all 1 1 comment reply Mk Utkarsh commented Mar 18, 2018 i edited by Mk Utkarsh Mar 18, 2018 reply Follow Share A lattice is distributive if it does not contain these two lattices 3 votes 3 votes Please log in or register to add a comment.
5 votes 5 votes in distributed lattice each element has atmost one complement in this lattice complement of 2 is 3 and 5. and hence it is not distributed lattice abhishekmehta4u answered Mar 17, 2018 abhishekmehta4u comment Share Follow See all 2 Comments See all 2 2 Comments reply Lakshman Bhaiya commented Mar 17, 2018 reply Follow Share please explain 1st options?? 0 votes 0 votes abhishekmehta4u commented Mar 17, 2018 reply Follow Share power set is always distributed lattice. there is no comparable element. simillarlly for 4 element 1 votes 1 votes Please log in or register to add a comment.
2 votes 2 votes i think option b). A propery that can be used to eliminate non-distributive lattice is that complement of element in lattice must be unique but here element 2,3,5 have 2 complements each. In option a) P(A) power set of a we will have 2^n elemets. in hasse diagram we'll have 16 nodes and 32 (n * 2^(n-1)) edges. therefore this is a boolean lattice. Hence distributed bcoz every boolean lattice is distributed and complemented. correct me if i'm wrong. Ananya Jaiswal 1 answered Mar 17, 2018 edited Mar 17, 2018 by Ananya Jaiswal 1 Ananya Jaiswal 1 comment Share Follow See all 14 Comments See all 14 14 Comments reply Show 11 previous comments Lakshman Bhaiya commented Mar 17, 2018 i edited by Lakshman Bhaiya Mar 17, 2018 reply Follow Share If lattice has to be distributed and complemented, so we say that boolean algebra.If it has to be boolean algebra then the number of elements in lattice diagram should be $2^{n}$ and number of edges should be $n.2^{n-1} $, Where n = number of elements. please check if I'm wrong? 1 votes 1 votes ankitgupta.1729 commented Mar 18, 2018 reply Follow Share @Lakshman ,u r right...A lattice is boolean algebra if and only if it is both complemented and distributed ..statement and its converse both r true here...A boolean algebra always have 2n elements and n.2n-1 edges.. 2 votes 2 votes Mk Utkarsh commented Mar 18, 2018 i edited by Mk Utkarsh Mar 18, 2018 reply Follow Share ankitgupta.1729 is right PS : you guys are answering so fast that i missed this question :p 0 votes 0 votes Please log in or register to add a comment.
0 votes 0 votes [P(A);⪯] where A = { a,b,c,d } In case of {a,b,c,d} the Hasse diagram of the Power Set will have no element with a complement, except the endpoints of the longest antichain. => Each element has at most one complement => Distributive lattice. [ {1,2,3,5,30} ; / ] Multiple complements of a single element would be found in the Hasse diagram for it. So, not distributive lattice. JashanArora answered Dec 19, 2019 JashanArora comment Share Follow See all 0 reply Please log in or register to add a comment.