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CSC165H1S Midterm
创建日期：2024-04-26 11:49:40
标签： CSC165H1S
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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MOEC0021 STA 106 ENGG 1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A 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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 IE5600 ECOS 3997 EDST2003 FIN 534 Elec4430 MATS3001 MGT6047 FINS5514 MIS602 ATS1279 ECMM445 2F CS262 EFIMM0097 FINM2002 ECMT3150 FINC5090 WACC1000 MECH4620 ECOS 2002 SEESGS79 STAT COMP3007 BUSB200F ACST8040 CAES9920 STAT0020 INFR11205 PHYS2111 FINM7402 STAT 428 SOFT3202 ME220 ACC-ACF2100 EC9410 CSE 120 COMP90042 SEE5211 ECON0048 BUSN2036 MKT2010 3SMFE4 EFIMM0117 ECON3034 MTH5120 EFIMM0120 X0 W19186 MATH 034 COMP2100 COMS4104 CS1210 MENG10005 ACCT 5906 ECMT1020 MGTS7607 MATH3090 CIV4100 CSYS5020 ELE2019 PHS4200 ACS6124 MSIN0225 PHAS0061 6CCS3BIM FINM7403 COMP8410 PY406 COMP2620 CS226 SDEV2004 COMP6080 BFF1001 HR410 MCEN30017 AMS3328 ML3 ELE8059 FINC3017 ACAD001 FTEC2101 ES480 3D ELECTENG 734 MTH731U MATH-UA 251 CSCI-UA 9473 A2 MSDM5058 FIT5047 COM6012 1FIN B386F MPHY202P ACCT6003 ECE 6913 HMB312H1 AC1025 MTH6105 Math6168 FINM 3003 CS24320 PHYS2134 ACCT3011 MECH1400 COMP30024 ENGGEN 121 COMP3721 MCD2080 ECON 7002 MAS439 COMP6216 ECON2011 M23367 CSC597 COMP6247 BRIEF QBUS6180 EENG22000 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