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Exam 2 - Introduction to Discrete Structures - Spring 2005 | MATH 455, Exams of Discrete Structures and Graph Theory

University of Massachusetts - AmherstDiscrete Structures and Graph Theory

Material Type: Exam; Class: Int Discrete Strctrs; Subject: Mathematics; University: University of Massachusetts - Amherst; Term: Spring 2005;

Typology: Exams

Pre 2010

Uploaded on 08/18/2009

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DEPARTMENT OF MATHEMATICS AND STATISTICS
UNIVERSITY OF MASSACHUSETTS
MATH 455.2 April 14, 2005
EXAM 2
Your Name:
Your student ID:
This exam paper consists of 7 questions. It has 6 pages. Show your work: all your an-
swers must be justified. When proving using induction make sure you include words
to describe what you are assuming and what you are trying to show.
No calculators, books or notes are allowed!
1. (20)
2. (20)
3. (15)
4. (15)
5. (10)
6. (15)
7. (15)
TOTAL (110)
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Download Exam 2 - Introduction to Discrete Structures - Spring 2005 | MATH 455 and more Exams Discrete Structures and Graph Theory in PDF only on Docsity!

DEPARTMENT OF MATHEMATICS AND STATISTICS

UNIVERSITY OF MASSACHUSETTS

MATH 455.2 April 14, 2005 EXAM 2

Your Name:

Your student ID:

This exam paper consists of 7 questions. It has 6 pages. Show your work: all your an- swers must be justified. When proving using induction make sure you include words to describe what you are assuming and what you are trying to show. No calculators, books or notes are allowed!

TOTAL (110)

  1. (20 pts) Consider the recurrence relation 4an = 4an− 1 − an− 2 , n ≥ 2.

(a) (10 pts) Find the general form of the solution.

(b) (10 pts) Give a formula for an valid for n ≥ 0 that satisfies the initial conditions a 0 = 3, a 1 = 2.

  1. (15 pts) How many one-to-one functions are there from the set of 5 elements to the set of 7 elements? Justify your answer, but do not attempt to write them all down.
  2. (15 pts) Give an example of a function f : Z → Z which is onto, but not one- to-one.
  1. (10 pts) Prove that 2 + 4 + 6 +... + 2n is Θ(n^2 ). You may use limits.
  2. (15 pts) Show that in a group of 3 people whose sum of ages is 67 years there will always be two people whose sum of ages is at least 45.