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Practice Final Exam - Introduction to Discrete Mathematics | MATH 240, Exams of Discrete Mathematics

Material Type: Exam; Class: Introduction to Discrete Mathematics; Subject: MATHEMATICS; University: University of Wisconsin - Madison; Term: Fall 2005;

Typology: Exams

Pre 2010

Uploaded on 09/02/2009

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MATH 240; FINAL EXAM, 150 points, 20 December, 2005 (R.A.Brualdi)
TOTAL SCORE (10 problems; plus one 15 point Bonus):
Name:
TA: Anders Hendrickson (circle time) Mon 12:05 Mon 1:20 Wed 12:05 Wed. 1:20
Do NOT compute factorials or binomial coefficients.
1. [10 points] Let Abe a set of 7 elements and let Bbe a set of 9 elements. What are:
The number of injective (one-to-one) functions ffrom Ato B:
The number of binary relations Rfrom Ato B:
2. [10 points] Let A={a, b, c}. Draw the diagram of the partially ordered (S, ) where S
is the power set of A. Then topologically sort the elements of this partially ordered set.
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Download Practice Final Exam - Introduction to Discrete Mathematics | MATH 240 and more Exams Discrete Mathematics in PDF only on Docsity!

MATH 240; FINAL EXAM, 150 points, 20 December, 2005 (R.A.Brualdi)

TOTAL SCORE (10 problems; plus one 15 point Bonus):

Name:

TA: Anders Hendrickson (circle time) Mon 12:05 Mon 1:20 Wed 12:05 Wed. 1:

Do NOT compute factorials or binomial coefficients.

  1. [10 points] Let A be a set of 7 elements and let B be a set of 9 elements. What are:
    • The number of injective (one-to-one) functions f from A to B:
    • The number of binary relations R from A to B:
  2. [10 points] Let A = {a, b, c}. Draw the diagram of the partially ordered (S, ⊆) where S is the power set of A. Then topologically sort the elements of this partially ordered set.
  1. [15 points] Twenty-four people are to be transported by 6 cars (a Toyota, a Subaru, a Ford, a Jeep, a Chevrolet, and a Chrysler) with 4 people per car.
    • In how many ways can the transportation be arranged?
    • If one person in each car is to have a designated driver, how many ways can the transportation be arranged?
  2. [10 points] For n ≥ 1, Let hn denote the number of ways for a person to climb a flight of n stairs when the person takes 1, 3, or 4 steps at a time. What is a recurrence relation for hn with initial conditions?
  1. [30 points] Consider the binary relations R on a set A defined below.
    • What are the three defining properties for R to be an equivalence relation?
    • What are the three defining properties for R to be a partial order?
    • Let R be the relation on A = {− 3 , − 2 , − 1 , 0 , 1 , 2 , 3 } defined by aRb provided |a| = |b| (absolute value). Is R an equivalence relation, total order, partial order, or none of these? (circle one) If an equivalence relation, how many different equivalence classes are there and what are they?
    • Let R be the relation on A = { 1 , 2 , 3 ,... , 20 } defined by aRb provided that a is a divisor of b. Is R an

equivalence relation, total order, partial order, or none of these? (circle one). If an equivalence relation, how many different equivalence classes are there and what are they?

  • Let R be the relation on the set A = { 1 , 2 , 3 ,... , 19 , 20 } of 20 elements defined by aRb provided that a ≡ b (mod) 6. Is R an

equivalence relation, total order, partial order, or none of these? (circle one)

If an equivalence relation, how many different equivalence classes are there and what are they?

  1. [10 points] Let A be a set of 4 elements, and let R be a binary relation on A.
    • If R is a symmetric relation and MR has 0’s and 1’s as shown:

   

   

, then

show what the other entries of MR are.

  • Let MR have 0’s and 1’s in positions as shown:

   

   

. The number of ways

to complete this to an anti-symmetric, binary relation on A equals:

  1. [20 points] Consider the poset (S, ≤) whose diagram is given below.

Determine

  • All maximal elements.
  • All minimal elements.
  • All the upper bounds of a and b, and then the LUB of a and b if it exists.
  • All the lower bounds of a and b, and then the GLB of a and b if it exists.
  1. [20 points]
    • The following is the postfix (postorder) form of a logical expression. What is its (unambiguous) usual form?

p q r ∨ ∧ q p∧ → p r ∧ ∨

  • What is the prefix (preorder) form of the algebraic expression

((x × y) + z) − ((z × (u + v)) + ((y × z) × (x + y)))

  • Prove by structural induction that n(T ) ≤ 2 h(T^ )+1^ − 1 a FBT T.