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数学代写 - Foundations of Computer Science (COMP9020)

时间：2020-12-06

Instructions:
• The exam consists of 10 questions for a total of 200 marks
• Solutions should be submitted as a pdf, maximum size 20Mb.
• Solutions should be submitted via give/webCMS. Multiple submissions are allowed, your last submission before the due date will be the submission marked.
• Deadline for submissions is 10am, Tuesday 8 December 2020 (AEDST). • Late submissions will be penalized at 25% (applied to raw mark) per 15 minutes
or part thereof.
• Existing written materials (i.e. passive resources) may be used, but should be
properly referenced. Obtaining, or trying to obtain solutions from active resources (e.g. social media, forums, other people) is a breach of the University’s
Plagiarism Policy and will be subject to disciplinary action as outlined in the
Student Code of Conduct.
• Unless explicitly stated, you should prove/explain/motivate your answers. You
are being assessed on your understanding and ability.
• Be careful with giving multiple or alternative answers. If you give multiple answers, then we will give you marks only for your worst answer, as this indicates
how well you understood the question.
• Number of pages in this exam paper: 8, including this cover page.
Question 1 (20 marks)
(a) Using only the functions | · |, b·c, d·e, div, %, basic arithmetic functions (+, ×, ,
and ÷), and numeric constants, define the following functions:
(i) f : R \ {0} → {−1, 1} given by f(x) =
1 if x < 0
1 if x > 0
(2 marks)
(ii) g : R → {0, 1} given by g(x) =
1 if x ∈ Z
0 if x ∈/ Z
(2 marks)
(iii) h : N → {0, 1} given by h(x) =
1 if 5|x
0 if 5 - x
(2 marks)
(iv) min : R × R → R given by min(x, y) = x if x ≤ y y if y < x
(2 marks)
(v) q : R → {0, 1} given by q(x) =
1 if x ∈ Z and 5|x
0 otherwise (2 marks)
(b) The following process leads to a rule for determining if a large number n is
divisible by 17:
• Remove the last digit, b, of n leaving a smaller number a. • Let n0 = a 5b. • Repeat with n0
in place of n.
So, for example, if n = 12345, then n0 = 1234 5 · 5 = 1209. Repeating would
create 120 5 · 9 = 75; 7 5 · 5 = 18; and so on.
(i) Prove that 17|n if and only if 17|n0. (6 marks)
This shows that we can repeat the process with n0
, and so on, until we reach a
“small enough” value that checking divisibility by 17 is straightforward.
(ii) Give, with justification, an asymptotic upper bound in terms of n, for the
number of times this process needs to be repeated before a negative n0
is
reached. (4 marks)
1
Question 2 (20 marks)
(a) Prove, or provide a counterexample to disprove, the following for all sets A, B:
(i) If there is a set X such that A ∩ X = B ∩ X then A = B. (4 marks)
(ii) If there is a set X such that A ⊕ X = B ⊕ X then A = B. (4 marks)
(iii) If there is a set X such that A ∩ X = B ∩ X and A ∪ X = B ∪ X then A = B.
(4 marks)
(b) Let Σ be a finite set. Prove, or provide a counterexample to disprove, the following
for all languages X,Y, Z ⊆ Σ∗:
(i) If XY = YZ then X∗Y = YZ∗. (4 marks)
(ii) If X∗Y = YZ∗
then XY = YZ. (4 marks)
Question 3 (20 marks)
Let (A, ) be a poset. Define f : A → Pow(A) as follows:
f(a) = {x : x ≤ a}.
(a) Prove that a b if, and only if, f(a) ⊆ f(b). (6 marks)
(b) Prove that f is injective. (4 marks)
(c) Prove that f is not surjective. (4 marks)
(d) Prove that if c = glb(a, b) exists, then f(c) = f(a) ∩ f(b). (6 marks)
Question 4 (20 marks)
(a) Prove that for all sets A, B, C there is a bijection between (A × B) × C and A × (B × C). (4 marks)
(b) Prove that f : S → T is injective if, and only if, for all functions g, h : U → S: f ◦ g = f ◦ h implies g = h
(8 marks)
2
(c) Order the following functions in increasing order of asymptotic complexity:
(I) n3√n
(II) n3
log(n)
(III) T(n) where T(0) = 1 and T(n) = T(n 3) + n3
(IV) T(n) where T(0) = 1 and T(n) = 6T(n/2) + n3
(V) (3
log n)2
(VI) 3(log n)2
Justify your ordering. (8 marks)
Question 5 (20 marks)
Consider the following code snippet, which takes an acyclic directed graph (which
does not contain loops) and returns a topological ordering of the vertices (i.e. vertex
order respects edge direction):
topSort(G):
If G has no vertices :
return an empty list
endif
Let v be any vertex of G
while v has in-degree > 0:
Let v0 be a vertex such that (v0, v) ∈ E
Let v = v0
end while
Let G0 be the graph that results from removing v and all incident edges
Let L = topSort(G0)
prepend L with v
return L
For this question you may assume that removing a vertex and all incident edges; or
prepending a vertex to a list, can be executed in O(1) time. Assume that accessing a
single matrix entry can be executed in O(1) time, but searching a row (column) of an
n × m matrix will take O(m) time (O(n) time, respectively).
Calculate the asymptotic running time of the above code snippet in terms of n: the
number of vertices of G, and m: the number of edges of G:
(a) When G is given as an n × n adjacency matrix. (6 marks)
(b) When G is given as an adjacency list. (6 marks)
(c) When G is given as an n × m incidence matrix. (8 marks)
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Question 6 (20 marks)
Recall from Assignment 2 the recursive definition of a binary tree:
• (B): An empty tree τ • (R): An ordered pair of binary trees (Tleft, Tright)
The height of a binary tree is defined to be the length of the longest path from the root
node (node with no predecessors) to a leaf. For convenience, we define the height of
τ to be 1. The example tree in Assignment 2 has height 2.
(a) Based on the recursive definition above, recursively define a function height(T)
that computes the height of a binary tree T. (4 marks)
A binary tree is balanced if it is the empty tree, or if the heights of its two subtrees
Tleft and Tright differ by at most 1. A binary tree is fully-balanced if every subtree is
balanced. The example tree in Assignment 2 is fully-balanced.
(b) Based on part (a) and the recursive definition above, recursively define a function
balanced(T) that returns an element of B indicating whether or not the binary tree
T is fully-balanced. (4 marks)
Recall from Assignment 2 the count function that counts the number of nodes in a
binary tree:
• count(τ) = 0 • count((Tleft, Tright)) = 1 + count(Tleft) + count(Tright)
(c) Let P(T) be the proposition that:
count(T) ≤ 2
height(T)+1
1.
Prove that P(T) holds for all binary trees T. (6 marks)
(d) Let Q(T) be the proposition that:
If balanced(T) then count(T) ≥ 2
height(T)
2 1.
Prove that Q(T) holds for all binary trees T. (6 marks)
4
Question 7 (20 marks)
(a) You are on a spaceship filled with crewmates and imposters. Imposters will
always lie, and crewmates will always tell the truth. The following statements are
made:
• Red says: Either myself and Blue are both crew, or we are both imposters.
• Orange says: At least one of Red and Green is an imposter.
• Green says: Red is an imposter.
• Blue says: Red is an imposter.
(i) Formulate this as a problem in propositional logic, defining the propositional variables and what they represent; any formulas that are relevant;
and the logical problem you are solving. (6 marks)
(ii) Use your answer from (i) to determine what possible combinations of crewmate/imposters there are that are consistent with this system. (6 marks)
(b) Prove, or provide a counterexample to disprove the following hold in all Boolean
Algebras:
(i) There is no x such that x = x0. (4 marks)
(ii) For all x, y, z: ((x ∧ y) ∨ (x0 ∧ z)) ∨ (y ∧ z) = (x ∧ y) ∨ (x0 ∧ z). (4 marks)
Question 8 (20 marks)
Prove or disprove:
(a) The degree sequence of a graph provides enough information to determine if it
has an Eulerian cycle. (4 marks)
(b) If a graph G has an even number of vertices and an Eulerian cycle, then it has an
even number of edges. (4 marks)
(c) If a graph G has an even number of vertices and an Eulerian cycle, then it has
chromatic number at most 2. (4 marks)
5
(d) The following graph:
• • • • • • • • • •
contains a subdivision of K4. (4 marks)
(e) K2,16 contains a subdivision of K4,4. (4 marks)
Question 9 (20 marks)
For n ∈ N with n > 0 let An = {1, 2, . . . , n}. Let E(n) denote the number of equivalence relations there are on An, and let F(n, k) denote the number of surjective functions there are from An to Ak.
(a) With justification, give a simple formula (using combinatorial functions defined
in lectures) for F(n, n). (4 marks)
(b) With justification, give a simple formula (using combinatorial functions defined
in lectures) for F(n, 2). (4 marks)
(c) With justification, give a recurrence equation for E(n) involving (one or more)
E(n0) where n0 < n. Remember to include a base case. (4 marks)
Hint: How many equivalence relations are there on An where the equivalence class of n
has exactly i elements?
(d) With justification, give a recurrence equation for F(n, k) involving (one or more)
F(n0, k0) where n0 < n and k0 ≤ k. Remember to include a base case. (4 marks)
Hint: Each f ∈ AAn k
can be viewed as f = g ∪ {(n, y)} where y ∈ Ak and g is a function
with domain An 1.
(e) Let E(n, k) denote the number of equivalence relations on An that have exactly k
equivalence classes. With justification, define E(n, k) in terms of F(n, k). (4 marks)
6
Question 10 (20 marks)
Prove, or provide a counterexample to disprove the following for all discrete probability distributions P, all events A and B, and all random variables X and Y:
(a) P(A \ B) ≥ P(A) P(B) (6 marks)
(b) If 0 < P(A) ≤ P(B) then P(A|B) ≤ P(B|A) (6 marks)
(c) P(X > E(X)) ≥ 12
(4 marks)
(d) If E(X) > E(Y) then P(X > Y) > 0 (4 marks)