Atmiya Institute of Technology & Science Discrete Mathematics by Meri Dedania Assistant Professor MCA department Atmiya Institute of Technology & Science Yogidham Gurukul Rajkot

Question Bank of POSET

Consider the relation of divisibility ‘|’ of the set Z of integers Consider the relation of divisibility ‘|’ of the set Z of integers. Is the relation an ordering of Z? Consider P(S) as the power set, i.e. the set of all subsets of a given set S.then investigate (P(S),) as a partially order set ,in which the symbol  denotes the relation of set inclusion Give an example of R which is both a partial ordering relation and equivalence relation

Let R be a binary relation on the set of all positive integers such that R = {(a,b) : a-b is an odd integer}. Investigate the relation R for reflexive , symmetric , antisymmetric ,transitive and also R is partial ordering relation? If a relation R is transitive , then prove that its inverse relation R-1 is also transitive. Let A={1,2,3,4} and consider the relation R={(1,1),(1,2),(1,3),(2,2),(2,4),(3,3),(3,4),(1, 4),(4,4)}.show that R is partial ordering and draw its Hasse diagram.

Draw the Hasse Diagram of the following POSETs <{1,2,3,4,6,9},|> <{3,6,12,36,72},|> <{2,3,4,9,12,18},|> <{2,3,5,30,60,120,180,360},|> Let A = {1,2,3,4,12} be defined by the partial order of divisibility on A, that is if a and b  A , a ≤ b iff a | b. draw the Hasse diagram of the POSET <A, ≤> Let S = {1,2,3} and A = P(S). Draw the Hasse diagram of the POSET with the partial order  (Inclusion)

Consider N = {1,2,3,……} be ordered by divisibility Consider N = {1,2,3,……} be ordered by divisibility. State whether each of the following subsets of N is linearly ordered {16,4,2} {3,2,15} {2,4,8,12} {6} {5,15,30} Let dm denote the set of divisor of m ordered by divisibility. Draw the Hasse diagrams of d15 d16 d17

Find two incomparable elements in the following POSETs <{1,2,4,6,8},|> Let D100 = {1,2,4,5,10,20,25,50,100} whose all the elements are divisors of 100. Let the relation ≤ be the relation | (divides) be a partial ordering on D100 Determine the GLB of B where B = {10,20} Determine the LUB of B where B = {10,20} Determine the GLB of B where B = {5,10,20,25} Determine the LUB of B where B = {5,10,20,25}

Draw the Hasse diagram for divisibility on the following Sets Draw the Hasse diagram for the “Less than or equal to” relation on the set A = {0,2,5,10,11,15} Draw the Hasse diagram for divisibility on the following Sets {1,2,3,4,5,6,7,8} {1,2,3,5,7,11,13} {1,2,3,6,12,24,36,48} {1,2,4,8,16,32,64}