COEN 231: Introduction to Discrete Mathematics
April 28th, 2023: 9:00am – 12:00pm
Time Allowed: 3 hours
Last Name: _________________________________First Name: ________________________________
Student Number (Concordia ID):_ ___________________________________________________
INSTRUCTIONS
1. This is a 180 minutes, Close Book test. A non-programmable calculator may be used, but no other material
will be allowed to be used including any blank paper.
2. Write your name and student number on your test paper in the space provided above. Don’t turn this page until
you are instructed to do so.
3. All questions are of valued based specified marks. Each part of a question has different value as defined at the
end of each part.
4. There are total of 10 Questions in this booklet.
5. Please complete all 10 questions on this booklet. Use the test booklet for all your work (including scratch work)
and you may use the back of pages. Do not tear out pages. Hand in the test booklet at the end of the
examination.
6. Make sure you number the solution parts according to the problem. You should show your work and explain
clearly what you are doing. Answers without explanation or shown work, where appropriate, will receive low
or zero points.
7. No questions are allowed during the exam. May the Force be with you!
Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
Maximum
10
10
10
10
10
10
10
10
10
10
Score
Question 1 (10 marks)
(i)
Determine if p → (¬q ∧ r) and ¬p ∨ ¬(r → q) are logically equivalent. [2 Marks]
(ii)
Prove that (p ∨ q) ∧ (¬p ∨ r) → (q ∨ r) is a tautology. [2 Marks]
(iii)
Consider P(x, y) being a predicate where the universe for the variables x and y is given by {1, 2, 3}. Suppose
P(1, 3), P(2, 1), P(2, 2), P(2, 3), P(3, 1), P(3, 2) are true, and P(x, y) is false otherwise. Identify the truth
of the following propositions? [2 marks]
a. ∀𝑥 ∃𝑦𝑃(𝑥, 𝑦)
b. ∀𝑦∃𝑥(𝑃(𝑥, 𝑦) → 𝑃(𝑦, 𝑥))
c. ∀𝑥∀𝑦(𝑥 ≠ 𝑦 → (𝑃(𝑥, 𝑦) ∨ 𝑃(𝑦, 𝑥)))
d. ∀𝑦∃𝑥(𝑥 ≤ 𝑦 ∧ 𝑃(𝑥, 𝑦))
(iv)
Translate the following statement into propositional logic and specify used propositions: In a newly
built Star-Base in the Andromeda Galaxy, you receive 80% discount on your first visit to fuel your
starship only if Gamora (name of a person) is on board. (Hint: Use simple propositions (no need for
predicates).) [4 marks]
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Question 2 (10 marks)
(i)
Prove that if x is rational and x ≠ 0, then 1/x is rational [2 marks]
(ii)
Give proof by contradiction of the following theorem: If x is an odd integer, then x + 2 is odd. [2 marks]
(iii)
Use rules of inference to show that “In a new Star-Base built at Knowhere (name of a planet), if no member
of the Guardians of the Galaxy (group of people) is present or if no member of Ravagers group is present,
then the shields will be up, and the Wine Festival will be held. If the Wine Festival is held, then the best
wine will be selected. Best wine was not selected” imply that “There is a member of the Guardians of the
Galaxy group present at the Star-Base.” [6 marks]
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Question 3 (10 marks)
(i)
What is the cardinality of each of these sets? [2 marks]
a. ∅
b. {∅}
c. {∅,{∅}}
d. {∅,{∅},{∅,{∅}}}.
(ii)
Let A, B, and C be sets. Show that (A − B) − C = (A − C) − (B − C). [3 marks]
(iii)
Suppose g: A → B and f: B → C where A = B = C = {1, 2, 3, 4}, g = {(1, 4),(2, 1),(3, 1),(4, 2)}, and
f = {(1, 3),(2, 2),(3, 4),(4, 2)}. [2 marks]
a. Find g ∘ g
b. Find g ∘ g ∘ g
(iv)
Suppose f: R → Z where 𝑓(𝑥) = ⌈2𝑥 − 1⌉ [3 marks]
a. If A = {x | 1 ≤ x ≤ 4}, find f(A).
b. If B = {3, 4, 5, 6, 7}, find f(B).
c. If C = {-9, -8}, find f−1(C).
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Question 4 (10 marks)
(i)
1 1
1
Let Ai = [- 𝑖 , 𝑖 ] and Bi = (1- 𝑖 , 1) for i = 1, 2, 3, … find the following items. [3 marks]
a. ⋃+∞
𝑖=1 𝐴𝑖
b. ⋂+∞
𝑖=1 𝐵𝑖
c. ⋃+∞
𝑖=1 𝐵𝑖
(ii)
Find a closed-form solution for ∑𝑛𝑗=8(𝑎𝑗−5 + 2 − 𝑎𝑗−7 ), where 𝑎0 , 𝑎1 , … , 𝑎𝑛 is a sequence of real numbers.
[3 marks]
(iii)
Find an explicit formula (function of n) that generates the following sequences: [4 Marks]
a. 5, 9, 13, 17, 21, . . . .
b.
0, 2, 0, 2, 0, 2, 0, . . . .
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Question 5 (10 marks)
(i)
Use the Principle of Mathematical Induction to prove that for all n ≥ 1 [5 marks]
𝑛
∑ 𝑖2𝑖 = (𝑛 − 1)2𝑛+1 + 2
𝑖=1
(ii)
Use the Principle of Mathematical Induction to prove that 2n + 3 ≤ 2n for all n ≥ 4. [5 marks]
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Question 6 (10 marks)
(i)
In a new Star-Base built in the Andromeda Galaxy, called Archi-Khan Base, there is a supreme council
consisting of 20 Guardians of the Galaxy (group of people) and 17 Ravagers. To discuss the fate of the
Rocket, they need to form a committee of size six. [6 Marks]
a. How many committees are possible if the committee must have three Guardians of the Galaxy and
three Ravagers?
b. How many committees are possible if the committee must have at least two Ravagers?
c. How many committees are possible if the committee must consist of all Guardians of the Galaxy or
all Ravagers?
(ii)
Consider all bit strings of length 12 [4 Marks]:
a. How many begin with 11 or end with 10?
b. How many have exactly four 1’s?
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Question 7 (10 marks)
𝑛
(i) Find closed-form solution of the following summation ∑𝑛𝑘=0 3𝑘 ( ) =?, when n is an integer. [3 Mark]
𝑘
(ii) Find the coefficient of 𝑥 8 in the expansion of (𝑥 2 + 2)13. [3 Marks]
(iii) Prove the following equality [4 Marks]
𝑛
2𝑛
( ) = 2 ( ) + 𝑛2
2
2
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Question 8 (10 marks)
(i) Determine whether the relation R is reflexive, symmetric, antisymmetric, and/or transitive [4 Marks]
a. The relation R on ℤ where 𝑎 2 = 𝑏2 .
b. The relation R on the set of all people where a R b means that a is at least as tall as b.
(ii) Suppose A is the set composed of all ordered pairs of positive integers. Let R be the relation defined on A
where (𝑎, 𝑏)𝑅(𝑐, 𝑑) means that 𝑎 + 𝑑 = 𝑏 + 𝑐. [4 Marks]
a. Prove that R is an equivalence relation.
b. Find [(2, 4)].
(iii) Suppose that 𝑅1 and 𝑅2 are equivalence relations on the set S. Determine whether 𝑅1 ∩ 𝑅2 is an
equivalence relation. [2 Marks]
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Question 9 (10 marks)
(i) Check different invariants, determine whether graphs A and B are isomorphic or not. [1 Marks]
A
B
(ii) Consider Graph C below: (a) Does it have an Euler circuit? (b) Does it have an Euler path? (c)
Does it have a Hamilton circuit? (d) Does it have a Hamilton path? [4 Marks]
C
(iii) Fill in the blanks in the following 5 items [2 Marks] :
a. The adjacency matrix for 𝐾𝑛 has ------ 1’s and ------ 0’s.
b. 𝐾𝑚,𝑛 has ------ edges and ------ vertices.
(iv) The complementary graph 𝐺̅ of a simple graph G has the same vertices as G. Two vertices
are adjacent in 𝐺̅ if and only if they are not adjacent in G. [4 Marks]
a. If G is a simple graph with 15 edges and 𝐺̅ has 13 edges, how many vertices does G
have?
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Question 10 (10 marks)
(i) Find all integers m and n for which the complete bipartite graph Km,n is a tree. [1 Mark]
(ii) Peter Quill (a person, a member of the Guardians of Galaxy) starts sending an invitation
letter to 8 Ravagers (name of a group of interstellar criminal. Each Ravager is asked to send
the letter out to 8 others, and each letter contains a list of the previous 7 Ravagers in the
sequence. Unless there are fewer than 7 names in the list, each person sends one Gold Coin
to the first person in this list, removes the name of this person from the list, moves up each
of the other six names one position, and inserts his name at the end of this list. If no Ravager
breaks the sequence and no Ravager receives more than one letter, how many Gold Coins
will Peter Quill (a person) in the sequence ultimately receives? [4 Marks]
(iii) Use Prim’s algorithm to find a minimal spanning tree for this weighted graph. [2 Marks]
(iv) First, draw all nonisomorphic trees with 5 vertices and then draw all nonisomorphic rooted
trees with 4 vertices. [3 Marks]
(v) Who is your favorite character in the Guardians of Galaxy (don’t miss the GoTG Vol. 3,
which is to be released on May 5th 2023!) [1 Bonus Mark!]
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Extra Page
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Extra Page
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Binomial theorem
𝑛
(𝑥 + 𝑦)𝑛 = ∑
𝑗=0
𝑛
( 𝑗 ) 𝑥 𝑛−𝑗 𝑦 𝑗
Pascal theorem
𝑛
𝑛
𝑛+1
(
)=(
) + ( ).
𝑘 −1
𝑘
𝑘
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