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CSC165H1S Midterm
Creation date：2024-04-26 11:49:40
Belonging category：计算机computer science
Midterm

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CSC165H1S Midterm 1, Version 1

No Aids Allowed
Name:
Student Number:
• This examination has 4 questions. There are a total of 7 pages, DOUBLE-SIDED.
• Final expressions in predicate logic must have negation symbols (¬) applied only to predicates or proposi-
tional variables, e.g., ¬p or ¬Prime(x).
• You may not define your own propositional operators, predicates, or sets unless asked for in the question.
• In your proofs, you may always use definitions we have covered in this course. However, you may not use
any external facts about these definitions unless they are given in the question.
• You may not use proofs by induction or contradiction on this midterm.
Take a deep breath.
This is your chance to show us
how much you’ve learned.
We WANT to give you the credit
that you’ve earned.
A number does not define you.
Good luck!
Question Grade Out of
Q1 7
Q2 8
Q3 5
Q4 5
Total 25
CSC165H1S , Winter 2020 Midterm 1, Version 1
1. [7 marks] Short answer questions. No justification is required for any parts of this question.
(a) [1 mark] Let S = {1, 3, 5, 7, 9}. Find a set S1 ⊆ N such that the following properties all hold:
|S1| = |S|+ 1 and ∀x ∈ S, ∃y ∈ S1, x− 1 = y
Solution
S1 = {0, 2, 4, 6, 8, 10} (10 could be replaced by any other natural number)
(b) [2 marks] Let T = {1, 3, 5, 7, 9}. Find sets T1 ⊆ T and T2 ⊆ T such that the following properties all hold:
T1 ⊆ T2 and |T1| > 1 and |T2 × T2| = |T1 × T | − 1
Solution
T1 is any two numbers from T2, and T2 is any three numbers from T , e.g., T1 = {1, 3} and T2 = {1, 3, 5}.
(c) [2 marks] Write down the truth table for the following expression in propositional logic. Intermediate
columns of the truth table are not required, but may be included.
(p⇒ ¬q)⇔ r
Solution
p q r (p⇒ ¬q)⇔ r
False False False False
False False True True
False True False False
False True True True
True False False False
True False True True
True True False True
True True True False
(d) [2 marks] Suppose we want to prove the following statement:
∀x ∈ N, (∃y ∈ N, P (x, y))⇒ (∃z ∈ N, Q(x, z))
Write the complete proof header for a proof, introducing all variables and assumptions. You may write
statements like “Let d = ” without filling in the blank. The last statement of your proof header should
be “We will prove that. . . ” where you clearly state what’s left to prove.
Please do not write below this line. There is extra space at the back of the test paper. Page 2/7
CSC165H1S , Winter 2020 Midterm 1, Version 1
Solution
Let x ∈ N.
Assume that there exists a y such that P (x, y) is true. Let z = .
We will prove that Q(x, z) is true.
Please do not write below this line. There is extra space at the back of the test paper. Page 3/7
CSC165H1S , Winter 2020 Midterm 1, Version 1
2. [8 marks] Translations. Let P be the set of all people, and suppose we define the following predicates:
• Doctor(x): “x is a doctor”, where x ∈ P
• Loves(x, y): “x loves y”, where x, y ∈ P
(Loves(x, y) does not mean the same thing as Loves(y, x))
Translate each of the following statements into predicate logic. No explanation is necessary. Do not define any of
your own predicates or sets. You may use the = and 6= symbols to compare whether two people are the same.
(a) [2 marks] Every doctor loves everyone.
Solution
∀p1 ∈ P, Doctor(p1)⇒
(∀p2 ∈ P, Loves(p1, p2))
(b) [2 marks] There is exactly one person loved by everyone.
Solution
∃p ∈ P, (∀p1 ∈ P, Loves(p1, p)) ∧ (∀q ∈ P, (∀p2 ∈ P, Loves(p2, q))⇒ p = q)
Or, ∃p ∈ P, ∀q ∈ P, (∀p2 ∈ P, Loves(p2, q))⇔ q = p
The following is tempting, but incorrect: ∃p ∈ P, ∀p1 ∈ P, Loves(p1, p)∧
(∀q ∈ P, Loves(p1, q)⇒ p = q).
By using the p1 in the last part, this says that q must equal p if any one person p1 loves q (rather than
all people), since the implication Loves(p1, q)⇒ p = q must be true for all p1 and q.
(c) [2 marks] If every person loves at least one doctor, then no one is a doctor.
Solution(∀p1 ∈ P, ∃p2 ∈ P, Loves(p1, p2) ∧Doctor(p2))⇒ (∀p3 ∈ P, ¬Doctor(p3))
(d) [2 marks] There exists a person who is not a doctor, who loves himself/herself, and only loves himself/herself.
Solution
∃p1 ∈ P, ¬Doctor(p1) ∧
(∀p2 ∈ P, Loves(p1, p2)⇔ p1 = p2)
Or, ∃p1 ∈ P, ¬Doctor(p1) ∧ Loves(p1, p1) ∧
(∀p2 ∈ P, p1 6= p2 ⇒ ¬Loves(p1, p2))
Please do not write below this line. There is extra space at the back of the test paper. Page 4/7
CSC165H1S , Winter 2020 Midterm 1, Version 1
3. [5 marks] A proof about numbers.
(a) [1 mark] Translate the following statement into predicate logic: “For every positive integer n, the sum of
the first (n2 − 1) positive integers is divisible by 2.”
Do not use the predicate |, but instead expand its definition in your statement. You may use summation
notation in your translation.
Solution
∀n ∈ Z+, ∃k ∈ Z,
n2−1∑
i=1
i = 2k
(b) [4 marks] Prove the statement from part (a) in the space below (you may also continue onto the next page).
Use a proof by cases based on whether n can be written as n = 2k or n = 2k + 1 for some integer k.
You may find the following formula helpful:
∀m ∈ N,
m∑
i=1
i =
m(m + 1)
2
Solution
Proof. Let n ∈ Z+. We’ll divide our proof into two cases.
Case 1: assume there exists a k1 ∈ Z such that n = 2k1.
Let k2 = k
2
1(n
2 − 1). We’ll prove that
n2−1∑
i=1
i = 2k2. We can calculate:
n2−1∑
i=1
i =
(n2 − 1)n2
2
(using the given formula)
=
(n2 − 1)(4k21)
2
(substituting n = 2k1)
= 2k21(n
2 − 1)
= 2k2 (by our choice of k2)
Case 2: assume there exists a k1 ∈ Z such that n = 2k1 + 1.
Please do not write below this line. There is extra space at the back of the test paper. Page 5/7
CSC165H1S , Winter 2020 Midterm 1, Version 1
Let k2 = (k
2
1 + k1)n
2. We’ll prove that
n2−1∑
i=1
i = 2k2. We can calculate:
n2−1∑
i=1
i =
(n2 − 1)n2
2
(using the given formula)
=
(4k21 + 4k1 + 1− 1)n2
2
(substituting n = 2k1)
= (2k21 + 2k1)n
2
= 2k2 (by our choice of k2)
Please do not write below this line. There is extra space at the back of the test paper. Page 6/7

4. [5 marks] Floors and disproofs.
Recall that the floor of a number x ∈ R, denoted bxc, is defined as the greatest integer that is less than or equal
to x. Consider the following statement:
There exists a k ∈ Z+ where every x ∈ Z+ satisfies x− (⌊√x⌋)2 ≤ k.
(a) [1 mark] Write the negation of this statement in predicate logic.
Solution
∀k ∈ Z+, ∃x ∈ Z+, x− (⌊√x⌋)2 > k
(b) [4 marks] Disprove the original statement by proving its negation in the space below.
Hint: for all positive integers a and b, if b2 ≤ a < (b + 1)2 then ⌊√a⌋ = b.
Solution
Proof. Let k ∈ Z+. Let x = k2 + 2k. We’ll prove that x− (⌊√x⌋)2 > k.
First, since k ≥ 1, we know that k2 + 2k > k2. Also, k2 + 2k < k2 + 2k + 1 = (k + 1)2, and so we can
conclude k2 ≤ x < (k + 1)2. The hint then tells us that ⌊√x⌋ = k.
So then we can calculate:
x− (⌊√x⌋)2 = x− k2 (since ⌊√x⌋ = k)
= k2 + 2k − k2 (since x = k2 + 2k)
= 2k
> k (since k ≥ 1)
Total Pages = 7. Total Marks = 25. Page 7/7
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ACCOUNTING BUSS1040 CHEM 181 MAT136H1 GY 6183 COMP3170 AGRI90089 PHYSICS 1002 ECON10003 IEOR E4004 ACS61012 MATH 40550 MKTG2113 TELE4653 ENSC1004 ECE 232E MK726 MTRX2700 SHEET 3 CSCA08 COMP30027 Math 2XX3 statistic MMAN3200 BISM7213 ECMM109 ECON 2022 FNCE 20005 GR5067 CSCB09 ME 3017 CS 5100 PHYS 2903 COMS 4771 COMP222 COMP 1046 CSC148H1 ISYS2038 FINS2615 OT5206 COMP2048 A6 MECH5770M MA826/20 CMPT 726 STA457 INFS6071 INFS588 integral ECON 216 IRDR 0017 MA323 PA 15213 PSYC2001 ENV222 Statistics ACCT5930 ACCT12 LAND2121 ME 5405 COMM1180 TELE3113 A7 ECE 568 ACCT2019 INFS1200 ENGG1340 CS 1400 ASST3 FINM7409 SWEN30006 INFO2222 FIN20150 SOSS1000 CSSE7231 FINS5513 COMS3200 CECN 702 ECON4060H MN3515 HW2 PSYA02H3 ECON 2112 COMP90051 CS 1652 MA30059 AF1605 MKTG7501 ECO-6003B COM4521 CTM ECON3350 COMP 3804 ELEC9762 BISM 7255 ST22 MATH20014 MNE3122 COMP343 MATH5165 ELE7029 ENGR30002 SEEM3500 ECON7350 RISK5002 FINM7405 COMP90059 MOS 3320B STAT 431 BISM7221 MATH3014 MTH783P EBA 35302 ACS133 BUS2206 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CMP7000A CMP-7038B Math 263 LAW3016 CS371 STAT 334 FIN0008 ACC3004 STAT 371 ​COMP1921 MTH201 ACCT2000 UN3412 CIV6000 MATHS 3021 MATE7014 ECON1 CIBC6035 LAW121G COMM 321 CSC8204R TTM2033 CSCI 2600 MAE 115 LNG302 CENV6175W1 ECON30019 CPS2015 BST969 DDES9015 AMATH242 CS 1151 N1592 ACCT600 EG503G SMM519 EC318 Ecstatistics EEE 471 MARK2051 Topology COM00037H ACCT3000 MTHM501 EG501V COM00055H BUSN9085 GGR273 MATH 2B ITD102 MA214 ECON0060 ECE3114 ECMM447 Mechanics CCT361H5S BSAN3210 WELF2001 Fin3001 RESP5001 STA 137 Linguistics 101 CSCI368 CMPSC497 LUBS3655 OMBA 5355 BUSM2562 ​BFF5902 CYBR7001 CULT2005 ECON 7720 MTRX3760 FSE598 ​LUE1001 MGSC 7100 CS3240 COSC7500 ACCT3091 INFOGR TRAN 2080 DESIGN CIVE215001 CIVE2360 CPSC 8430 FNCE2003 Stoichiometry INFS6023 Quantitative Method Math 21D ECE5179 Design ​ECON8022 Chemical DMS 213 ADM1370 BFF5902 BFF2701 AGRI10051 AMN400 ECON7322 ​FINC5090 ​MATH1052 ​SciComp Elec4622 ​FINS5516 ENG 101 MMM143 CS431CA2 BSAN2201 EBSC7070 ES9v9-10 PLIN0009 DATA 8 MATH 101 BCPM0097 ​MAT 237 BISM2201 EEE8099 Physics 11 ECE2238B Biostat 234 ​ECON30009 INF10025 CHEM252 CCMA4008 CIVE 3007 STAT0031 ELEC ENG 3108 MGMT90140 CHEM 252 STAT3016 MMM095 ​MMGT6016 Legal STAT 311 ELEC9764 COMP 370 TCS 3053 EECS 1021 FIT1051 C/C++ EE3C COMP2003J COM398 COMP51915 Simulator FIN42330 8PRO102 PGD ELEC5160 XJCO1921 COMP5123M INFS 2042 CHE1148H ECA5313 IN3329 Microsoft Modeling B15 2TT COMP3687 COMP2432 DMS2030 KAN4TSF CFKG CSE 5322 CSE 445 STATS 3DA3 CHC5223 COMP1048 EL3105 EECS 3221 CS231 ME 4434 ITP4510 ITP4501 CSCI 5708 Controller EECE 5699 CPEN 231L CPN programs CSC8016 MATU9D2 ENGN4213 2. SVM. (10 points) Consider an SVM classifier using RBF kernel, and finish the following python code to search for the optimum kernel parameter γ. You are required to calculate the RBF function and search yourself, instead of using functions in Scikit-Learn (sklearn). def parameter search(gamma list ,Xtr,ytr,Xts,yts ,...): """ Parameters gamma list: A list of RBF kernel parameter gamma to search for Xtr: Training data, shape (ntr,nrow,ncol) ytr: Training labels, shape (ntr,) Xts: Test data, shape (nts,nrow,ncol) yts: Test labels, shape (nts,) ... Add other parameters as needed Returns: gamma optimum: The gamma with minimum error on Xts, yts """ ... return gamma optimum
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