Discrete Mathematics Exam II
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Questions and Answers

What is the primary rule regarding resources allowed during the exam?

  • Students may use notes as references.
  • Calculators are encouraged for problem-solving.
  • The exam is closed-book and no outside resources are allowed. (correct)
  • Communication with peers is permissible.

    • How should a student handle a question they find difficult during the exam?

  • Skip it and move on without marking it.
  • Mark it and return to it later. (correct)
  • Attempt to solve it even if it takes a long time.
  • Spend the entire time focusing on that question.

    • What is the total number of scoring points available in the exam?

  • 90 points, with no possibility of extra credit.
  • 100 points, plus up to 10 points in extra credit. (correct)
  • 110 points, totaling both the main exam and extra credit.
  • 100 points, with an additional 5 points for extra credit.

    • How should students indicate they need extra space for their answers?

    <p>Write 'continued' on the extra sheet.</p> Signup and view all the answers

    What should students be cautious about when reading the exam questions?

    <p>New combinations or variations of questions could appear.</p> Signup and view all the answers

    What is the correct expression for the inductive step in proving P(n) by mathematical induction?

    <p>∀k(P(k) =⇒ P(k + 1)) for all positive integers k</p> Signup and view all the answers

    If a task can be accomplished in either n1 ways or n2 ways, what is the total number of ways to complete the task?

    <p>n1 + n2 by the sum rule</p> Signup and view all the answers

    How many different ways can a 5-letter password be chosen from 52 letters without repeating any letter?

    <p>52!/47!</p> Signup and view all the answers

    How many one-to-one (injective) functions can be formed from a set A to itself if |A| = 5?

    <p>5!</p> Signup and view all the answers

    How many bit strings of length 10 do not contain a consecutive sequence of 9 '1's?

    <p>29 − 2</p> Signup and view all the answers

    What is the number of ways to choose an unordered team of 40 students from 89 registered students?

    <p>C(89, 40)</p> Signup and view all the answers

    In how many different ways can a committee of 3 members be formed from a group of 10 members?

    <p>C(10, 3)</p> Signup and view all the answers

    What is the product rule in counting?

    <p>It applies when there are multiple stages and options at each stage.</p> Signup and view all the answers

    What is the correct expression for S1 + S2?

    <p>$\sum_{j=0}^{n} (2 \cdot 3^{j} + 3 \cdot 2^{j})$</p> Signup and view all the answers

    Which type of progression does the sum of terms represent?

    <p>Geometric progression</p> Signup and view all the answers

    If set A is countably infinite, which statement is true?

    <p>All of the above</p> Signup and view all the answers

    What is the relationship between the cardinality of the sets Z and N?

    <p>|Z| = |N|, and both are countably infinite</p> Signup and view all the answers

    What do the symbols in the expressions for S1 and S2 represent?

    <p>A series of geometric terms</p> Signup and view all the answers

    How can one define the countable infinity of set A?

    <p>It can be matched directly to the natural numbers</p> Signup and view all the answers

    What is the key difference between countably infinite and uncountably infinite sets?

    <p>Countably infinite sets can be listed, whereas uncountably infinite cannot</p> Signup and view all the answers

    In the context of set theory, what can be inferred about the cardinality of the natural numbers N?

    <p>It is always greater than the cardinality of any finite set</p> Signup and view all the answers

    Which formula correctly represents the sum of the first n positive integers?

    <p>$n(n + 1)/2$</p> Signup and view all the answers

    How many different arrangements can three people have when seated around a circular table?

    <p>6</p> Signup and view all the answers

    What is the sum of the geometric series $1 + 2 + 4 + ... + 2^n$ for any non-negative integer n?

    <p>$2^{n+1} - 1$</p> Signup and view all the answers

    What is a valid statement about the set of positive odd integers?

    <p>It is countably infinite.</p> Signup and view all the answers

    What is the result of $1 + 2 + 2^2 + ... + 2^n$?

    <p>$2^{n+1} - 1$</p> Signup and view all the answers

    Which concept is essential for proving the sum of the first n positive integers using induction?

    <p>Base case validation</p> Signup and view all the answers

    Which of the following formulas is used for the sum of a finite geometric series?

    <p>$S_n = rac{ar^{n+1} - a}{r - 1}$ for r ̸= 1</p> Signup and view all the answers

    How many bit strings of length 10 start with six 0s or end with five 0s?

    <p>192</p> Signup and view all the answers

    Study Notes

    Discrete Mathematics Exam II - Critical Information

    • Exam Type: Closed-book
    • Materials Allowed: No notes, internet, calculators, programs, or communication devices.
    • Work Style: Work efficiently. If a problem is difficult, move on and come back to it later.
    • Question Format: Read questions carefully. Expect variations and new combinations of previous homework questions.
    • Continued Work: Use the back of previous pages or blank pages for additional space. Indicate the continuation with "continued" or "cont" and the problem number.
    • Scoring: Exam is out of 100 points total, plus possible extra credit of 10 points.
    • Multiple Choice: Multiple choice questions have only one correct answer. Numbers may differ from other exams.

    Discrete Mathematics Exam II - Exam Questions

    • Question 1 (6 pts.): Σ (3⁰ + 2⁰) for integers n > 0

      • Options (choose one): a) Σ (3⁰ + 2⁰) = (3+2)^n b) Σ (3⁰ + 2⁰) = Σ(3⁰) + Σ(2⁰) c) Σ (3⁰ + 2⁰) = (3 + 2) d) none of the above

    • Question 2 (6 pts.): Sum of terms (23^2 + 32^2)

      • Options (choose one): a) arithmetic progression b) geometric progression c) both a and b d) none of the above

    • Question 3 (6 pts.): If set A is countably infinite

      • Options (choose one): a) injection from A to set of natural numbers b) surjection from set of natural numbers to A c) bijection between A and set of natural numbers d) all of the above e) none of the above

    • Question 4 (6 pts.): Cardinality

      • Options (choose one): a) If Z is uncountably infinite and N is countably infinite, |Z| > |N| b) If Z and N are both uncountably infinite, |Z| > |N| c) If Z and N are countably infinite, |Z| = |N| d) none of the above

    • Question 5 (6 pts.): Mathematical Induction

      • Options (choose one): a) To prove P(n) by induction, prove k(P(k) => P(k+1)) b) P(1) => P(k) => P(k+1) => P(n) c) P(1) => for all k(P(k) => P(k+1)) => P(n) d) (P(1) ∧ (P(k) ∧ P(k+1)) => P(n))

    • Question 6 (6 pts.): Combining task possibilities. Given n₁ and n₂ task ways.

      • Options (choose one): a) n₁ + n₂ by sum rule b) n₁ * n₂ by product rule c) n₁! * n₂! by product rule d) None of the above

    • Question 7 (6 pts.): Determine the number of possible 5-letter passwords using the English alphabet (case-sensitive). No letters can be repeated.

    • Question 8 (6 pts.): Number of one-to-one functions from set A to itself, where |A| = 5.

    • Question 9 (6 pts.): Bit strings of length 10 with no consecutive nines.

    • Question 10 (6 pts.): Number of ways to choose an unordered team of 40 students from 89 students.

    Discrete Mathematics Exam II - Induction Proof

    • Question 2.1: Prove by induction that 1+2+3+…+n = n(n+1)/2 for any integer n ≥ 1.
    • Question 2.2: Prove by induction that 1+2^1+2^2+……+2^n = 2^(n+1)-1 for any integer n is ≥ 0

    Discrete Mathematics Exam II - Counting

    • Question 3.1: Is the set of positive odd integers countably infinite? Provide a function proof.
    • Question 3.2: How many ways are there to seat three people around a circular table with three chairs?
    • Question 3.3: How many bit strings of length 10 either begin with six 0s or end with five 0s (or both)?

    Discrete Mathematics Exam II - Extra Credit

    • Question 4: Prove the formula for the sum of a finite geometric progression. Use mathematical induction.

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    Prepare for the Discrete Mathematics Exam II with this focused quiz. It covers critical information about the exam format, scoring, and types of questions you will encounter. Improve your problem-solving skills and readiness for varied question styles in this closed-book assessment.

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