CMPSCI 190DM: A Mathematical Foundation for Informatics

Solutions to Second Midterm Exam

David Mix Barrington

15 November 2013

Directions:

• Answer the problems on the exam pages.
• There are four problems for 100 total points. Actual scale was A = 90, C = 60.
• If you need extra space use the back of a page.
• No books, notes, calculators, or collaboration.

  Q1: 20 points
  Q2: 30 points
  Q3: 25 points
  Q4: 25 points
Total: 100 points

• Question 1 (20): Briefly identify the following terms or concepts (4 points each):
  • (a) a onto function

    An onto function f from A to B is one where every member of B is equal to f(x) for at least one element x of A.

  • (b) an transitive relation

    A transitive relation R ⊆ A × A is a binary relation where whenever R(x, y) and R(y, z) are both true (for any elements x, y, and z of A, possibly equal), the statement R(x, z) is also true.

  • (c) an equivalence relation

    An equivalence relation is the relation {(x, y): x and y are in the same set of P} for some partition P. Equivalently, it is any relation that is reflexive, symmetric, and transitive.

  • (d) the power set of a set

    The power set P(X) of a set X is the set whose members are all subsets of X. That is, P(x) = {Y: Y ⊆ X}.

  • (e) the function P(n, r)

    P(n, r) is the number of permutations of size r taken from an n-element set. A permutation is a sequence that has no elements repeated.

• Question 2 (30): Let A, B, and C be any three sets. Assume that the statements "A ⊆ B" and "A ∩ C = ∅" are both true.
  • (a, 10) Draw a Venn diagram indicating the relationships among A, B, and C.

    The diagram has circles for B and C that are intersecting, and a circle for A that is inside the portion of B that is not part of C.

  • (b, 10) Consider the statement "(B ∩ C) - A = ∅". Can this statement be proved either true or false from the information given? If it can, do so.

    It might be either true or false. If all three sets are empty, it is true, and the given statements A ⊆ B and A ∩ C = ∅ are also true. If B = C = {x} and A = ∅, the two given statements are both true but (B ∩ C) - A = ∅ is false.

  • (c, 10) Consider the statement "A - (B ∪ C) = ∅". Can this statement be proved either true or false from the information given? If it can, do so.

    The statement is true given the two premises A ⊆ B and A ∩ C = ∅. If there were an element x of A = (B ∪ C), it would be in A. Then x ∈ B, because A ⊆ B, and X ∈ B ∪ C by the definition of union. So x is not in A - (B ∪ C), a contradiction.

• Question 3 (25): Let S be the set of dogs {Ace, Biscuit, Cardie, Duncan, Ebony}. Let R be the following binary relation on S: R = {(x, y): dog x has fewer letters in his/her name than dog y does}.
  • (a, 10) Draw the diagram of the relation R.

    A has arrows to all four other nodes, C and D have arrows only to B, and E has arrows to C, D, and B.

  • (b, 5) Is R reflexive? Explain your answer.

    It is not. R(x, x) is always false as a dog cannot have fewer letters in its name than it has in its name. R is in fact irreflexive.

  • (c, 5) Is R anti-symmetric? Explain your answer.

    It is antisymmetric. We must show that if R(x, y) and R(y, x) are both true, then x = y. But "R(x, y) ∧ R(y, x)" is impossible, as x cannot have both a shorter and a longer name than y. So the implication is true vacuously -- there could never be a counterexample with R(x, y) ∧ R(y, x) true but x = y false.

  • (d, 5) Is R transitive? Explain your answer.

    R is transitive. If R(x, y) and R(y, z) are both true, then x's name has fewer letters than y's and y's has fewer than z's, so x's name has fewer letters than z's and thus R(x, z) is true.

• Question 4 (25): This question concerns three-letter words over the 26-letter alphabet {A, B, C,..., Z}. We consider all three-letter strings, not just those that are words in English. (Remember that an arithmetic expression need not be evaluated for a full-credit answer.)
  • (a, 5) How many three-letter words are there in all?

    By the Product Rule, there are 26³ = 17576 elements of the three-fold direct product of the alphabet.

  • (b, 10) For this problem, a vowel is a member of the set {A, E, I, O, U} and a consonant is any letter that is not a vowel. How many of the three-letter words have consonants in the first and third positions and a vowel in the second position?

    There are 21 choices for the first letter, 5 for the second, and 21 for the third, so we have 21 × 5 × 21 = 2205 total words of this type.

  • (c, 10) How many of the three-letter words counted in part (a) do not have the same letter twice (or more) in a row?

    There are at least three good methods to solve this problem:

    Method 1: Take the 26³ total strings, subtract the 26² strings with first and second letter matching, subtract the 26² with second and third letter matching, and add back in the 26 strings with all three letters matching. 26 × (26² - 26 - 26 + 1) = 26(26 - 1)² = 26(25)² = 16250.

    Method 2: 26 choices for the first letter, 25 for the second (any letter but the first) and 25 choices for the third letter (any letter but the second), 26 × 25 × 25 = 16250 total.

    Method 3: 26 × 25 × 24 with all three letters different, plus 26 × 25 with first and third letters the same but second letter different, total 26 × 25 × (24 + 1) = 26 × 25 × 25 = 16250.

  Last modified 12 December 2013