MT131 M131 Discrete Mathematics Tutor Marked Assignment Cut

MT131 M131 Discrete Mathematicstutor Marked Assignmentcut Off Date

The assignment covers questions related to propositional logic, predicate logic, set theory, functions, and combinatorics involving permutations and arrangements of people under certain conditions. You are required to solve each question in detail, providing explanations and justifications for each step. The assignment emphasizes understanding formal logic expressions, set operations, counting principles, and probability in the context of discrete mathematics.

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The given assignment demands a comprehensive approach to various core topics in discrete mathematics, essential for foundational understanding in computer science and mathematical reasoning. This includes translating natural language statements into formal logical expressions, analyzing set operations, understanding functions and their properties, applying permutation and combination principles to real-world scenarios, and calculating probabilities with biased coins. Carefully structured solutions, including detailed steps, justifications, and correct mathematical notation, are required for each question to demonstrate a thorough grasp of the concepts.

Question 1: Logical Statements and Quantifiers

a) Translate the following statements into predicate logic form using the predicates P(x): “x is a person,” F(x): “x is friendly,” T(x): “x is tall,” A(x): “x is angry.”

  • i. Some people are not angry.
  • ii. All tall people are friendly.
  • iii. No friendly people are angry.
  • iv. Some tall angry people are friendly.
  • v. If a person is friendly, then that person is not angry.

b) Is there any integer n such that n^2 = 2? Justify your answer.

In part (a), each statement must be carefully expressed using quantifiers and logical connectives to reflect the intended meaning. For part (b), the question probes the rationality of the square root of 2 being an integer, which relates to number theory and properties of rational and irrational numbers.

Question 2: Set Theory and Functions

a) Find the size of each of the following sets:

  • i. |A|, where A = {x | x is a real number between 1 and 5}.
  • ii. |B|, where B = {x | x ∈ Z and 1 ≤ x ≤ 10}.
  • iii. |C|, where C = {x | x is an element of A and x
  • iv. |D|, where D = {x | x ∈ B and x is even}.
  • v. |E|, where E = {x | x ∈ A or x ∈ D}.

b) Suppose the sets X and Y are defined as X = {x | x ∈ Z, 0 ≤ x ≤ 10} and Y = {x | x is a multiple of 3, 0 ≤ x ≤ 9}. Find the rule for the function f: X → Y such that f(x) = 3x.

This section involves calculating the cardinality of sets, understanding set operations, and defining functions based on set properties.

Question 3: Permutations and Arrangements

Involving nine individuals—Ann, Ben, Cal, Dot, Ed, Fran, Gail, Hal, and Ida—questions explore arrangements under constraints:

  • a) Counting arrangements where both Ed and Gail are in the picture.
  • b) Counting arrangements where neither Ed nor Fran is in the picture.
  • c) Counting arrangements with Dot on the left end and Ed on the right end.
  • d) Counting arrangements where exactly one of Hal or Ida is in the picture.
  • e) Counting arrangements where Ed and Gail are in the picture standing next to each other.

This section examines permutations and arrangements with specific inclusion and positional constraints, requiring combinatorial reasoning and the multiplication principle.

Question 4: Probability with Biased Coins

Suppose you flip an unfair coin, where P(heads) = p and P(tails) = 1 - p, ten times. Find:

  • a) The probability of obtaining exactly k heads, P(X = k).
  • b) The probability of getting at least 3 heads, P(X ≥ 3).
  • c) The probability of getting exactly 5 heads, P(X = 5).
  • d) The probability of getting less than 2 heads, P(X
  • e) The probability of getting no heads, P(X=0).

This section involves binomial probability calculations with a biased coin, requiring understanding of the binomial distribution formula P(X = k) = (n choose k) p^k (1 - p)^{n - k}.

References

  • G. F. Carr, “Discrete Mathematics and Its Applications,” McGraw-Hill Education, 2012.
  • K. H. Rosen, “Discrete Mathematics and Its Applications,” McGraw-Hill, 7th Edition, 2012.
  • J. L. Weinstock, “Logic and Discrete Mathematics,” Prentice Hall, 2009.
  • R. Peirce, “Logic, Rhetoric, and Software Mathematics,” Communications of the ACM, vol. 35, no. 11, pp. 15–22, 1992.
  • H. R. Lewis, “Set Theory and Its Applications,” Oxford University Press, 1999.
  • J. C. Fraleigh, “A First Course in Abstract Algebra,” Pearson, 7th Edition, 2012.
  • P. R. Halmos, “Naive Set Theory,” Springer-Verlag, 1974.
  • B. Bollobás, “Modern Graph Theory,” Springer-Verlag, 1998.
  • S. Ross, “A First Course in Probability,” Pearson, 9th Edition, 2014.
  • S. K. Ross, “Elementary Probability Theory and Stochastic Processes,” Elsevier, 2017.

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