SFWR ENG 2DM3 Discrete Mathematics and Logic Problem Set 3 7 Questions

SFWR ENG 2DM3
Discrete Mathematics and Logic
Problem Set 3
7 Questions
Released: October 27, 2014
Due date: November 26, 2014
This problem set covers chapters 7-11 of Discrete Mathematics and Its Applications, 7th ed., McGraw Hill, by
Kenneth H. Rosen. Some of these questions deviate slightly from the book. Please answer the questions given here;
the source is provided mainly for reference.
** justify your answers **
1. (Page 452, exercise 36)
Which is more likely: rolling a total of 8 when two dice are rolled or rolling a total of 8 when three
dice are rolled?
2. The probability that it rains tonight is 0.2. The probability that it rains tonight and I stay home is
0.1, The probability that it rains tonight or I stay home is 0.4.
a) What is the probability that I stay home?
b) Does my decision to stay home depend on rain?
3. We want to tile a 2 × n rectangular board with 2 × 1 rectangular dominoes. Let an be the number of
such tilings of a 2 × n board.
a) Determine the values of a1 and a2 .
b) Find a recurrence relation for an in terms of an−1 and an−2 .
4. Let R = {(n, n2 )|n ∈ Z} be a relation on Z. Prove or disprove R is reflexive, irreflexive, symmetric,
antisymmetric, and transitive. (Note that a relation R on a set A is irreflexive if for every a ∈ A,
(a, a) ∈
/ R. That is, R is irreflexive if no element in A is related to itself.).
5. Exercise 16, page 615
Let R be the relation on the set of ordered pairs of positive integers such that ((a, b), (c, d)) ∈ R if and
only if ad = bc. Show that R is an equivalence relation.
6. Determine if the following graphs are isomorphic to each other. Either exhibit an isomorphism or
provide a rigorous argument that none exists.
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7. (Page 756, exercise 22)
A chain letter starts when a person sends a letter to five others. Each person who receives the letter
either sends it to five other people who have never received it or does not send it to anyone. Suppose
that 10,000 people send out the letter before the chain ends and that no one receives more than one
letter. How many people receive the letter, and how many do not send it out?
• Your solution must reach your TA by or before the due date
• You or your reliable friend must give your work to your TA hand-to-hand or deliver at class to your
instructor
• You have to sign your work; your signature certifies that the assignment is your work
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