Discrete Structures for Computer Science Section – 3 - Tutorial – 4 (11 th Sep 2013)

Discrete Structures for Computer ScienceSection – 3 - Tutorial – 4 (11th Sep 2013) JagatSeshChalla

Problem - 1 Minimal – 1, Maximal – 18,24 Upper Bounds of 4 & 6 - 12,24 Least Upper Bound of 4 & 6 - 12 Lower Bounds of 4 & 6 – 2 Greatest Lower Bound of 4 & 6 – 12 Chains – {1,2,4,8,24}, {1,2,6}, {2}, {1}, {1,2,6,12,24}, {1,3,,6,18}, etc. Anti Chains – {2,3}, {4,6,9}, {1}, {2}, {8,9,12}, {6,9,8}, etc. Length of Longest chain is 5 {1,2,4,8,24}. So by Theorem 4.1 the whole Partial order can be partitioned into 5 disjoint anti chains. They are – {1}, {2,3}, {4,6,9}, {8,12,18} & {24}. 24 8 18 12 4 9 6 2 3 1

Problem - 2 Which among these are lattices? Is divisibility on set of Natural Numbers a lattice?

Generating Functions Generating function of the sequence is For example the generating function of the sequence 1,1,1,… is Similarly the generating function of the sequence 1, -1, 1, -1, … (-1)r,… is

Generating Functions Things to remember

Generating Functions

Problem - 3 Through Partial Fractions, find the coefficient for the generating function Solution: Through Partial Fractions we have

Problem - 3 So from this we get rth term in the sequence, i.e.

Problem - 4 Through Partial Fractions, find the coefficient for the generating function Solution:

Problem - 4 So coeff of is

Problem - 5 Write the generating function of the sequence Solution:

Problem - 10 Find the coefficient of in Solution:

Problem - 10 Coefficient of is

Problem - 11 Find coefficient of in Solution:

Problem - 11 Coefficient of is

Problem - 12 Using generating function find the number of ways to put 30 balls in to five numbered boxes where each box contains at least 3 balls and at most 7 balls. Solution: The generating function is: Now the problem boils down to finding coefficient of in the above generating function.

Problem - 13 Through generating functions find the number of ways to find the number of ways to select 10 balls from a large pile of black, red and blue balls if the selection has at most 2 red balls. Solution: The generating function is: The problem boils down to finding the coefficient of in the above generating function.

SURPRISE

Quiz – 1 Problem – 1: (3 marks) Given the set A = {1,2,3, 4,6,8,12, 24} of positive divisors of 24, define a partial order on A by a M b  a is a multiple of b. Draw Hasse Diagram for this Poset. Problem – 2: (2 marks) Is the relation ⊆ of set inclusion equivalence relation? Prove or disprove it.

Solutions to Quiz 1. 2. Property of Symmetry Fails. So its not an equivalence relation. 1 3 2 6 4 12 8 24

Q & A

Discrete Structures for Computer Science Section – 3 - Tutorial – 4 (11 th Sep 2013)