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MAST20026 set of real
Creation date：2024-05-10 17:10:58
Belonging category：数学mathematics
set of real

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MAST20026

In this exam paper, R is the set of real numbers, R+ is the set of positive real
numbers, Q is the set of rational numbers, Z is the set of integers, and N is the set
of positive integers.
Question 1 (7 marks)
Let P be the statement
P : (q _ ⇠ r) =) ((⇠ q) ^ (⇠ r))
where q, r are primitive statements.
(a) Construct a truth table for the statement P .
(b) If P is true, what can you conclude about q? What can you conclude about r? Explain
briefly.
Question 2 (10 marks)
(a) (i) Express the following statement using only mathematical notation:
Any integer which is not a perfect square is either a multiple of 2 or a multiple of 3.
(Note: A number is a perfect square if it is the square of an integer.)
(ii) Is the statement in (i) true? Explain briefly, with mathematical justification.
(b) Consider the statement
8x, y 2 R x < y =) 9z 2 R x < z < y
(i) Give an equivalent statement using only mathematical notation but without using a
conditional.
(ii) Negate the statement, and simplify by eliminating negation symbols ‘⇠’ as much as
possible.
Question 3 (6 marks)
Prove that the product of any irrational number and any non-zero rational number is irrational.
Page 2 of 5 pages
MAST20026 Semester 1, 2017
Question 4 (6 marks)
Listed below are axioms of the real numbers.
A1. 8x, y 2 R, x+ y 2 R
A2. 8x, y, w, z 2 R (x = w^y = z) =) x+y = w+z
A3. 8x, y 2 R, x+ y = y + x
A4. 8x, y, z 2 R, (x+ y) + z = x+ (y + z)
A5. 9y 2 R, 8x 2 R, x+ y = x
A6. 8x 2 R, 9y 2 R, x+ y = 0
O1. 8x, y 2 R, exactly one of the following is true:
x < y, y < x, x = y.
O2. 8x, y, z 2 R, if x < y and y < z then x < z
M1. 8x, y 2 R, xy 2 R
M2. 8x, y, w, z 2 R (x = w ^ y = z) =) xy = wz
M3. 8x, y 2 R, xy = yx
M4. 8x, y, z 2 R, (xy)z = x(yz)
M5. 9y 2 R(8x 2 R, xy = x)
M6. 8x 2 R \ {0}, (9y 2 R, xy = 1)
D1. 8x, y, z 2 R, x(y + z) = xy + xz
AO. 8x, y, z 2 R, x < y =) x+ z < y + z
MO. 8x, y, z 2 R, (x < y ^ 0 < z) =) xz < yz
Using only these axioms (and rules of logic), prove that
8x, y, z 2 R
⇣
zx = zy ^ z 6= 0 =) x = y⌘
Question 5 (16 marks)
(a) For a set A ✓ R, state the definition of
(i) ‘A is bounded above’
(ii) ‘A is bounded’
(b) Recall that a set A ✓ Q is a cut if it satisfies properties (I)-(III) below:
(I) A 6= ; and A 6= Q
(II) for all x, y 2 Q, if x 2 A and y < x then y 2 A
(III) if x 2 A then there exists some y 2 A with y > x.
(i) Prove that every cut is bounded above.
(ii) Prove that every cut is unbounded.
(c) What can you conclude about the supremum and infimum of a cut?
Question 6 (12 marks)
(a) State the ✏-M definition of lim
n!1 fn = L, where (fn) is a sequence and L 2 R.
(b) Let fn =
n 6
3n2 + 1
. Prove using the definition that lim
n!1 fn = 0.
(c) Is the sequence (fn) from part (b) a Cauchy sequence? Explain briefly. A complete proof
is not required.
(d) Give an example of each of the following, or explain why they do not exist.
(i) A bounded, non-monotonic sequence
(ii) A divergent, bounded, monotonic sequence
(iii) An unbounded convergent sequence
If you are giving an example, it is sucient to just write down your example; you do
not need to prove that it satisfies the given properties. If you claim that no such example
exists, then quote a relevant fact from this subject to support your claim.
Page 3 of 5 pages
MAST20026 Semester 1, 2017
Question 7 (14 marks)
Let f : R! R be given by f(x) =
(
x2 x 2 Q
x2 x /2 Q .
(a) At which point(s) x 2 R is f continuous? Prove that f is continuous at these points using
the ✏- definition.
(b) Prove that f is not continuous at all other points.
Question 8 (6 marks)
Use the Intermediate Value Theorem to prove that the equation ex + x = 4 has a real solution.
Question 9 (6 marks)
Recall that
• U(f, P ) and L(f, P ) denote the upper and lower Riemann sums, respectively, of f over
the partition P ;
• U(f) = inf{U(f, P ) : P 2 eP} and L(f) = sup{L(f, P ) : P 2 eP}
where f : [a, b] ! R is a function, P is a partition of [a, b], and eP is the set of all partitions of
[a, b].
(a) Explain briefly why U(f) and L(f) exist for any bounded function f : [a, b]! R.
(b) Let f : [0, 1] ! R be given by f(x) = x2. Does L(f) = U(f)? If so, what is its value?
Justify your answer briefly.
Question 10 (8 marks)
Use appropriate tests to determine:
(a) if the series
1X
n=1
2n + 1
n22n + n
converges.
(b) if the integral
Z 1
1
e2x
21 dx exists as an improper Riemann integral.
Page 4 of 5 pages
MAST20026 Semester 1, 2017
Question 11 (18 marks)
Let f : R+ ! R be given by f(x) = log(x).
(a) Prove by induction that f (n)(x) =
(1)n1(n 1)!
xn
for all n 2 N.
Note: f (n)(x) denotes the n’th derivative of f . You may use the usual di↵erentiation rules
for ddx(log x) and
d
dx(x
↵) without further proof.
(b) Recall that the Taylor series of a function g about a point x = a is
1X
n=0
g(n)(a)
n!
(x a)n.
Verify that the Taylor series of f(x) = log(x) about x = 1 is
1X
n=1
(1)n1
n
(x 1)n.
(c) Determine the interval of convergence of the Taylor series for f(x). Justify your conclu-
sions with appropriate convergence tests or facts from this subject.
(d) At which point(s) x 2 R does the Taylor series converge absolutely? At which point(s)
does it converge conditionally? Justify your answer.
(e) Could the function f be represented as a Fourier series? Briefly justify your answer.
End of Exam—Total Available Marks = 109
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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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