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Midterm1

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School
University of California, Berkeley**We aren't endorsed by this school
Course
MATH 55
Subject
Mathematics
Date
Dec 16, 2024
Pages
10
Uploaded by ColonelRain373
Math 55 First Midterm Exam, Prof. SrivastavaOctober 3, 2024, 5:10pm–6:30pm, Wheeler 150.Name:SID:GSI:Name of the student to your left:Name of the student to your right:Instructions:Write all answers clearly in the provided space. This exam includes somespace for scratch work which will not be graded.Do not unstaple the exam.Write yourname and SID on every page. Show your work — numerical answers without justificationwill be considered suspicious and will not be given full credit.Write proofs in completeEnglish sentences. Calculators, phones, cheat sheets, textbooks, and your own scratch paperare not allowed.UC Berkeley Honor Code:As a member of the UC Berkeley community, I act withhonesty, integrity, and respect for others.Sign here:QuestionPoints11226364858610Total:50Do not turn over this page until your instructor tells you to do so.
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Name and SID:1. Select true or false (by filling in the circle) for each of the following. There is no need toprovide an explanation.(a) (3 points) The compound proposition((p∨q)∧(¬p∨r))→q∨ris a tautology.⃝True⃝False(b) (3 points) The compound propositions(∀xP(x))→(∀xQ(x))and∀x(P(x)→Q(x))are logically equivalent.⃝True⃝False(c) (3 points) The cardinality of the interval (0,1) ={x∈R: 0< x <1}is equal tothe cardinality of the interval (0,2) ={x∈R: 0< x <2}.⃝True⃝False(d) (3 points) IfA, B, Care finite sets andf:A→B, g:B→Care functions suchthatg◦f:A→Cis onto, then bothfandgmust be onto.⃝True⃝False[Scratch Space Below]Math 55 Midterm 1Page 2 of 1010/3/2024
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Name and SID:2. (6 points) Prove or disprove: ifAandBare sets, thenP(A)∩P(B) =P(A∩B),whereP(A) ={S:S⊆A}denotes the power set ofA.[Scratch Space]Math 55 Midterm 1Page 3 of 1010/3/2024
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Name and SID:3. (6 points) Prove that ifAandBare countably infinite sets, thenA∪Bis countablyinfinite.[Scratch Space]Math 55 Midterm 1Page 4 of 1010/3/2024
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Name and SID:4.(a) (6 points) Prove that ifxandyare rational numbers withx < ythenx+y−x√2is irrational. You may not use any properties of rational numbers other than thedefinition. You may use that√2 is irrational.(b) (2 points) Prove that ifxandyare rational numbers withx < y, then there is anirrational numberzsuch thatx < z < y.Math 55 Midterm 1Page 5 of 1010/3/2024
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Name and SID:[Scratch Paper 1]Math 55 Midterm 1Page 6 of 1010/3/2024
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Name and SID:5.(a) (4 points) Prove or disprove: ifa, m >1 are integers withgcd(a, m) = 1, then thefunctionf:{0,1, . . . , m−1} → {0,1, . . . , m−1}defined byf(x) = (ax+ 3)modmis a bijection.(b) (4 points) Prove that ifa, m >2 are integers withgcd(a, m) = 1, then the functiong:{0,1, . . . , m−1} → {0,1, . . . , m−1}defined byg(x) = (ax2+ 3)modmisnota bijection.Math 55 Midterm 1Page 7 of 1010/3/2024
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Name and SID:6.(a) (4 points) Use the extended Euclidean algorithm to findgcd(4,31) and express itas a linear combination of 4 and 31 with integer coefficients.(b) (3 points) Use your solution to (a) to find the set of all integersxsuch that4x≡333(mod 31).Math 55 Midterm 1Page 8 of 1010/3/2024
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Name and SID:(c) (3 points) Find the remainder 4333mod31.[Scratch Space 2]Math 55 Midterm 1Page 9 of 1010/3/2024
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Name and SID:[Scratch Paper 2]Math 55 Midterm 1Page 10 of 1010/3/2024
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