Download For Marker`s use only. Page Mark Maximum 2 6 3 11 4 6 5 7 6 8 7 7

RYERSON UNIVERSITY
DEPARTMENT OF MATHEMATICS
MTH 314
SAMPLE EXAM
Winter, 2010
Total marks: 60
Time allowed: 3 hours.
Instructions:
• Verify that your paper contains 15 questions on 10
pages, the last of which is a table of set identities.
• An aid sheet, consisting of both sides of one 8.5”
x 11” sheet and conforming to the specifications
stated on the course web page may be used. Put
your name and student number on your aid-sheet
and hand it in with your exam paper for bonus
marks.
• No other aids are allowed.
• Fill in your name, student number, and signature
in the spaces provided above. Keep your Ryerson
photo identification card visible on your desk.
• Electronic devices such as, calculators, PDAs, headsets, cell-phones, pagers, portable tape, disk and
MP3 players must be turned off and kept inaccessible during the test. Except for religious reasons
hats may not be worn. You must not pass any item
(including a pen, pencil or eraser) to another student.
• Marks will be deducted if a solution is messy, illegible or disorganized.
• You must show all your work. The correct answer
alone may be worth nothing. Please write only in
this booklet. Use of scrap paper or additional enclosures is not allowed. If you need more space, continue on the back of the page, directing the marker
where the solution continues with a bold sign.
For Marker’s use only.
Page
Mark Maximum
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BONUS
Total
60
Your student number:
1. Use mathematical induction to prove that for all positive integers n:
1! + 2! + 3! + · · · + n! < 3n!.
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2. Prove or disprove that for all real numbers x and y
⌈x⌉ · ⌊y⌋ = ⌊x⌋ · ⌈y⌉.
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3. A certain graph G has 105 vertices, 100 edges, and no nontrivial circuits. How many connected
components does it have?
Explain your answer.
4. (a) If a graph has vertices of degrees 1, 1, 1, 2, 3, how many edges does it have?
(b) A graph with six vertices has degrees 1, 1, 2, 2, 3, 3. Either draw such a graph or explain
why no such graph exists.
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5. In this question Z− denotes the set of negative integers. Indicate which of the following relationships is true and which is false. WARNING: marks will be deducted for wrong answers.
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(a) Q ∪ Z = R ∩ Q
(d) Q − R ⊆ Z − R
(c) Z+ ⊆ Q
(f) Q ∈ P(R)
(b) (Z+ × Z− ) ∪ (Z− × Z+ ) = Z2
(e) (Q − Z+ ) ∩ (R − Z− ) = ∅
The True statements are:
The False statements are:
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6. Use the “element method” (i.e. showing that the sets on either side of a set equation have
exactly the same elements) to prove that for all sets A and B
A ∪ (A ∩ B) = A.
Your student number:
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7. Use the properties listed in the attached table (Theorem 5.2.2 in the textbook) to construct an
algebraic proof that for every pair of subsets A and B of a universal set U :
(A ∪ B c ) ∩ (A ∪ B) = A.
Be sure to justify every step in your proof with a reference to an identity in the table.
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8. Draw the transition diagram for a finite state automaton with input alphabet = {0, 1} that
accepts exactly those strings that start with 00 or 10.
Your student number:
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9. Show that Z+ and Z+ × Z+ have the same cardinality.
10. Use Euclid’s algorithm to find the greatest common divisor of 10509 and 1023.
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Your student number:
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11. Recall that a rational real number is one that can be expressed as the quotient of two integers.
Prove or disprove that for every pair of real numbers a and b, if a × b is rational then a is
rational or b is rational.
12. Find a regular expression that denotes the language consisting of all strings over the alphabet
{0, 1} which do not begin with 11.
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13. Let G be the graph given below.
V2
V1 e 1
e3
e4
V4
e8
e6
e5
V5
e 10
e11
e 12
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e16
e 15
e 19
V3
e7
V6
e9
e13
V7
e118
V9
e2
e 14
V8
e 17
V10 e 20
V11
(a) Determine whether G has a Hamiltonian circuit. If it does have a Hamiltonian circuit,
find such a circuit. Otherwise, explain why you can be 100% sure that it does not have a
Hamiltonian circuit.
(b) Determine whether G has a Euler circuit. If it does have a Euler circuit, find such a circuit.
Otherwise, explain why you can be 100% sure that it does not have a Euler circuit.
14. Draw the graph that has the following adjacency matrix:

1 0 1
A =  0 0 1 .
1 1 0

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Compute B = A2 . What is the significance of the value of b3 3 ?
Your student number:
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15. Do ONLY ONE of (a) or (b).
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(a) Give a rigorous proof that there can be no finite state automaton which accepts exactly
those strings over the alphabet {0, 1} which have an equal number of 0’s and 1’s.
(b) Define the sequence a0 , a1 , a2 , . . . , as follows:
a0 = 1, a1 = 1 and, an+1 = an−1 + an for all integers n ≥ 1.
Use strong mathematical induction to prove that for all integers n ≥ 0, we have

Ã
√ !n+1
√ !n+1 Ã
1− 5
1+ 5
1
.
−
an = √ 
2
2
5
Your student number:
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