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MAST20026 set of real
创建日期：2024-05-10 17:10:58
所属分类：数学mathematics
标签： MAST20026
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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