Discrete Mathematics Answers

Helen wants to buy a bunch of flowers. There are five types of flowers available, namely roses, chrysanthemums, lilies, carnations and tulips. How many possible bunches of 16 flowers can she choose if the bunch must contain no more than two lilies, but any number of the types of flowers.

Use generating function to prove the identity

n

∑

k=o



r

k

 s

n−k



=



r +s

n



Using Boolean algebra simplify the statement ¬(𝑟 → 𝑠) → (¬𝑟) 

1 Let a relation R be represented by the following matrix MR =

1000001 1111000  1 0 1 1 0 1 0 

MR=1 0 0 1 0 0 0

 1 1 0 1 0 0 0  0 1 0 1 1 0 1 0111100

 

Determine whether R is

(a) reflexive

(b) irreflexive

(c) symmetric

(d) asymmetric

(e) antisymmetric

(f) transitive

GIVE REASONS for your answer.

2 Consider the following relation R on A where A = {1, 2, 3, 4,5}

aRb ⇔ ab < min(a, b)

For example, 2R4 since 24 = 12 and min(2,4) = 2 and 12 < 2.

(a) Draw the digraph of R

(b) Give a path of length 2 from 3, if any

(c) Give the domain and range of R.

(d) Determine R

6 a) Show, using the pigeonhole principle, that in any set of 5 integers, at least two have the same remainder when divided by 4.

(b) Use the extended pigeonhole principle to show that there are at least 3 ways of choosing 2 different numbers from 2, 3, 4, 5, 6, 7, 8, 9 so that all choices have the same sum.

7 Decide for each of the following relations whether or not it is an equivalence relation. Give full reasons. If it is an equivalence relation, give the equivalence classes.

(a) Leta,b∈Z. DefineaRbifandonlyif ab ∈Z (4)

(b) Let a and b be integers. Define aRb if and only if 3|(a − b) (In other words R is the

congruence modulo 3 relation

4 What is the probability that an arrangement of a, b, c, e, f, g begins and ends with a vowel?

5 Helen wants to buy a bunch of flowers. There are five types of flowers available, namely roses, chrysanthemums, lilies, carnations and tulips. How many possible bunches of 16 flowers can she choose if the bunch must contains no more than two lilies, but any number of the other types of flowers?

(a) How many code words over a, b, c, d of length 20 contain exactly 10 a’s?

(b) How many contain exactly 10 a’s and 5b’s.

There are 30 pupils in a class. (a) You make a row of 7 pupils. In how many ways can this be done? (b) You divide the class into two groups of 15 each. In how many ways can this be done? (c) You give each of the 30 pupils one type of cooldrink from 5 different types of cooldrink. In how many ways can this be done? (d) Ignoring who gets which cooldrink, how many different cooldrink combinations are possible if you choose 30 cooldrinks from 5 types?

Write notes on how to find cartesian products of sets and in you concluding page, site examples on the applications of set theory to solving real world business problems

(A∪B)

c

(A∪B)c