Combinatorics | Number Theory Answers

Find the transitive closure of the relation using Warshall’s algorithm R={(1,2),(2,1),(2,3),(3,4),(4,1)} on the set {1,2,3,4}.

Is the `divides’ relation on the set of positive integers
(a) reflexive? (b) symmetric? (c) transitive? (d) an equivalence relation?
Justify your answers.

For the poset ({2,4,6,9,12,18,27,36,48,60,72},|) having the `divides’ relation, find
(a) the maximal elements
(b) the minimal elements
(c) All upper bounds of {2,9}
(d) The least upper bound of {2,9} if it exists.

What is a partially ordered set? Prove that the inclusion relation `is a subset of’ is a partial ordering on the power set of a finite set S.

Use generating functions to solve the recurrence relation a_k=5a_(k-1)-6a_(k-2) with the initial conditions a_0=6 and a_1=30.

Use generating functions to solve the recurrence relation a_k=3a_(k-1)+4^(k-1) with the initial condition a_0=1.

Find the solution of the recurrence relation a_n=4a_(n-1)-4a_(n-2)+(n+1).2^n.

Find the solution of the recurrence relation a_n=2a_(n-1)+〖3.2〗^n.

Find all solutions of the recurrence relation a_n=5a_(n-1)-6a_(n-2)+2^n+3n.

Find the general form of the solutions of the recurrence relation a_n=8a_(n-2)-16a_(n-4).