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Student Number: A/U/T*
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NATIONAL UNIVERSITY OF SINGAPORE
FACULTY OF SCIENCE
SEMESTER 1 EXAMINATION 2012-2013
MA1100 Fundamental Concepts of Mathematics
November/December 2012 Time allowed: 2 hours
INSTRUCTIONS TO CANDIDATES
1. Write down your matriculation/student
number neatly in the space provided above.
This booklet (and only this booklet) will be col-
lected at the end of the examination. Do not insert
any loose pages in the booklet.
2. This examination paper contains a total of EIGHT
(8) questions and comprises NINETEEN (19)
printed pages.
3. Answer ALL questions. Write your answers and
working in the spaces provided inside the booklet
following each question.
4. Total marks for this exam is 100. The marks for
each question are indicated at the beginning of the
question.
5. Candidates may use calculators. However, they
should lay out systematically the various steps in
the calculations.
Examiner’s Use Only
Questions Marks
1
2
3
4
5
6
7
8
Total
PAGE 2 MA1100
Question 1 [10 marks]
(a) Use mathematical induction to prove that 1 + 4n < 2n for all integers n ≥ 5.
(b) A sequence is defined by
a1 = 2, a2 = 4, an+2 = 5an+1 − 6an for all n ≥ 1.
Use a version of mathematical induction to prove that an = 2n for all n ∈ Z+.
Show your working below and on the next page.
(a) Let P (n) be 1 + 4n < 2n.
Basis step P (5) :
1 + 4× 5 = 21 < 25 = 32.
So P (5) is true.
Inductive step P (k)→ P (k + 1) (for k ≥ 5):
Given 1 + 4k < 2k.
Then 1 + 4k + 4 < 2k + 4
This implies 1 + 4(k + 1) < 2k + 2k (since k ≥ 5, so 4 < 2k.)
Hence 1 + 4(k + 1) < 2(2k) = 2k+1.
So P (k)→ P (k + 1) is true for all k ≥ 5.
By principle of mathematical induction, P (n) is true for all n ≥ 5.
(b) Let P (n) be an = 2n.
Basis step P (1) and P (2):
We have a1 = 2 = 21, a2 = 4 = 22.
So P (1) and P (2) are true.
Inductive step P (k − 1) ∧ P (k)→ P (k + 1) (for k ≥ 2):
ak+1 = 5ak − 6ak−1
= 5(2k)− 6(2k−1)
= 2k−1(5× 2− 6)
= 2k−1 × 22 = 2k+1.
By strong principle of mathematical induction, P (n) is true for all n ≥ 1.
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PAGE 3 MA1100
(More working spaces for Question 1)
Continue on page 18-19 if you need more space. Please indicate clearly.
· · · − 4−
PAGE 4 MA1100
Question 2 [15 marks]
(a) Define sets A and B as follows:
A = {n ∈ Z | n = 8r − 3 for some integer r} and
B = {m ∈ Z | m = 4s + 1 for some integer s}.
Prove A ⊆ B using element method.
(b) Let A,B,C be three subsets of a universal set. Use algebra of sets to prove that
(A−B) ∩ (A− C) = A− (B ∪ C).
State the properties (laws) that you use for your steps.
(c) Let An =
{1
n,
2
n, · · · , n− 1
n
}for integer n ≥ 2.
Use element method to prove that
∞⋃n=2
An = {q ∈ Q | 0 < q < 1}.
Show your working below and on the next page.
(a) Let x ∈ A.
Then x = 8r − 3 = 4(2r)− 4 + 1 = 4(2r − 1) + 1.
So x = 4s + 1 with s = 2r − 1 ∈ Z.
So x ∈ B.
Hence A ⊆ B.
(b)
LHS = (A−B) ∩ (A− C)
= (A ∩Bc) ∩ (A ∩ Cc) set difference
= (A ∩ A) ∩ (Bc ∩ Cc) associative/commutative
= A ∩ (B ∪ C)c idempotent/de Morgan
= A− (B ∪ C) set difference
= RHS
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PAGE 5 MA1100
(More working spaces for Question 2)
(c) First part (⊆):
Let x ∈∞⋃n=2
An.
Then x ∈ An for some integer n ≥ 2.
So x =m
nfor some integer m with 1 ≤ m ≤ n− 1.
Therefore x ∈ Q and 0 < x < 1.
i.e., x ∈ {q ∈ Q | 0 < q < 1}.
So we conclude∞⋃n=2
An ⊆ {q ∈ Q | 0 < q < 1}.
Second part (⊇):
Let x ∈ {q ∈ Q | 0 < q < 1}.
Then we can write x =m
nfor some integers m,n ∈ Z+ with 0 < m < n. (This is
possible for n ≥ 2.)
So x ∈ An for some integer n ≥ 2.
i.e.,x ∈∞⋃n=2
An.
So we conclude {q ∈ Q | 0 < q < 1} ⊆∞⋃n=2
An.
Continue on page 18-19 if you need more space. Please indicate clearly.
· · · − 6−
PAGE 6 MA1100
Question 3 [10 marks]
Define a function f : R− {0} → R by the formula f(x) =x + 3
x.
(i) Prove that f is one-to-one (injective).
(ii) Replace the codomain of f with a subset of R so that f becomes a bijection. Justify
your answer.
(iii) Write down the inverse function of the bijection in part (ii).
Show your working below and on the next page.
(i) Let x, y ∈ R− {0} such that f(x) = f(y):
⇒ x + 3
x=
y + 3
y⇒ xy + 3y = xy + 3x
⇒ 3y = 3x
⇒ x = y
So f is one-to-one.
(ii) Replace the codomain by R− {1}.
It is sufficient to show f : R−{0} → R−{1} is onto (in order that f is a bijection).
Let k ∈ R− {1}. Want to show there exists x ∈ R− {0} such that f(x) = k.
Take x =3
k − 1. Note that this represents a non-zero real number. Then
f
(3
k − 1
)=
3k−1 + 3
3k−1
=3k
3= k.
So f is an onto function, and hence a bijection (in view of part (i)).
(iii) f−1 : R− {1} → R− {0} with f−1(x) =3
x− 1.
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PAGE 7 MA1100
(More working spaces for Question 3)
Continue on page 18-19 if you need more space. Please indicate clearly.
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PAGE 8 MA1100
Question 4 [15 marks]
(a) Let a and b be integers such that a ≡ 5 mod 7 and b ≡ 4 mod 7.
(i) Find integers s, t where 0 ≤ s, t < 7 such that a + b ≡ s mod 7 and
ab ≡ t mod 7.
(ii) Is there an integer c such that ac ≡ 2 mod 14? Justify your answer.
(b) Use congruence modulo to compute the remainder of 18199 when it is divided by 65.
(c) Let a, b and n > 1 be integers. Prove that if m > 1 is a divisor of n and a ≡ b
mod n, then a ≡ b mod m.
Show your working below and on the next page.
(a) (i) a + b ≡ 5 + 4 ≡ 2 mod 7. (i.e. s = 2)
ab ≡ 5× 4 ≡ 6 mod 7. (i.e. t = 6)
(ii) Yes, we can take c = 6.
a ≡ 5 mod 7 ⇒ 3a ≡ 15 ≡ 1 mod 7
⇒ 2(3a) ≡ 2(1) mod 2(7)
⇒ 6a ≡ 2 mod 14
(b) First we note that 182 = 324 ≡ −1 mod 65.
We shall write 199 = 2× 99 + 1.
So 18199 = 182×99+1 = (182)99 × 18 ≡ (−1)99 × 18 ≡ −18 ≡ 47 mod 65.
So the remainder of 18199 is 47 when divided by 65.
(c) We are given m | n (1).
We also have a ≡ b mod n which implies n | (a− b) (2).
By (1) and (2), and the transitivity of divisibility, we have m | (a− b).
By definition of congruence, this implies a ≡ b mod m.
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PAGE 9 MA1100
(More working spaces for Question 4)
Continue on page 18-19 if you need more space. Please indicate clearly.
· · · − 10−
PAGE 10 MA1100
Question 5 [15 marks]
(a) Let A = {1, 2, 3, 4}. Let R be the relation on A given by a R b if and only if a + b
is even.
(i) Express R as an ordered pair representation. (List all the ordered pairs in R.)
(ii) Show that R is an equivalence relation.
(iii) Find the distinct equivalence classes of R.
(iv) Is it possible to remove one ordered pair from R so that the resulting relation
on A is reflexive, symmetric but not transitive? Justify your answer.
(b) Determine whether the following relation S on Z is reflexive, symmetric and tran-
sitive? Justify your answers.
S = {(a, b) ∈ Z× Z | (a− b)2 < 9}.
Show your working below and on the next page.
(a) (i) R = {(1, 1), (1, 3), (2, 2), (2, 4), (3, 1), (3, 3), (4, 2), (4, 4)}.
(ii) Reflexive: For all x ∈ A, x + x = 2x is even. So (x, x) ∈ R.
Symmetric: For all x, y ∈ A, x+y is even implies y+x is even. In other words,
(x, y) ∈ R→ (y, x) ∈ R.
Transitive: For all x, y, z ∈ A, if x+y and y+z are even, then (x+y)+(y+z) =
x + 2y + z is even. Since 2y is even, so x + z is also even. Hence (x, z) ∈ R
We conclude that R is an equivalent relation.
(iii) The distinct equivalent classes are [1] = {1, 3} and [2] = {2, 4}.
(iv) No. If we remove (a, a) for any a ∈ A, the resulting relation will not be
reflexive.
If we remove (a, b) for any two distinct a, b ∈ A, then we will have (b, a) in the
resulting relation but not (a, b), and hence the relation is not symmetric.
· · · − 11−
PAGE 11 MA1100
(More working spaces for Question 5)
(b) S is reflexive:
For all a ∈ Z, (a− a)2 = 0 < 9→ (a, a) ∈ S.
S is symmetric:
For all a, b ∈ Z, (a− b)2 = (b− a)2.
So (a− b)2 < 9→ (b− a)2 < 9.
So (a, b) ∈ S → (b, a) ∈ S.
S is not transitive:
For example, a = 1, b = 3, c = 5:
Then (a− b)2 = 4 < 9 so (a, b) ∈ S.
(b− c)2 = 4 < 9 so (b, c) ∈ S.
(a− c)2 = 16 > 9 so (a, c) 6∈ S.
Continue on page 18-19 if you need more space. Please indicate clearly.
· · · − 12−
PAGE 12 MA1100
Question 6 [10 marks]
Determine whether the following sets are countable or uncountable. Justify your answers.
(a) A = {n ∈ Z | n is a square free number}.
(b) B = {x ∈ R | x =√n for some integer n }.
(c) C = [0, 1]−{
1,1
2,1
3, . . . ,
1
n, . . .
}.
Show your working below and on the next page.
(a) A is countable.
A is a subset of Z which is countable. So A is countable.
(b) B is countable.
We define a function f : B → Znonneg by
f(x) = x2.
For any x =√n ∈ B, f(
√n) = n, which is a non-negative integer. So f is well-
defined.
For any x, y ∈ B, x =√n and y =
√m for some non-negative integers n,m. If
f(x) = f(y), then n = m which implies√n =
√m. So x = y, and hence f is
injective.
For any n ∈ Znonneg, let x =√n ∈ B. Then f(x) = f(
√n) = n. So f is surjective.
Since f is a bijection and Znonneg is contable, so B is countable.
(c) C is uncountable.
Note that C ∪{
1, 12, 13, . . . , 1
n, . . .
}= [0, 1].
Suppose C is countable.
Since{
1, 12, 13, . . . , 1
n, . . .
}is countable (being a subset of Q+), then [0, 1] is the union
of two countable sets, which is countable. This is contradiction to the fact that [0, 1]
is uncountable.
So we conclude that C is uncountable.
· · · − 13−
PAGE 13 MA1100
(More working spaces for Question 6)
Continue on page 18-19 if you need more space. Please indicate clearly.
· · · − 14−
PAGE 14 MA1100
Question 7 [15 marks]
Let S be the set of all finite strings in 0’s and 1’s where the left most digit is 1.
(a) Define a function g : S → Z as follows:
for each string s in S,
g(s) = the number of 1’s in s minus the number of 0’s in s.
(i) What is g(101011)? g(100100)?
(ii) Is g one-to-one? Prove or give a counterexample.
(iii) Is g onto? Prove or give a counterexample.
(b) Define a function h : S → Z+ as follows:
for each string s = anan−1 . . . a1a0 in S,
h(s) = an × 2n + an−1 × 2n−1 + · · ·+ a1 × 2 + a0.
Is h a bijection? Justify your answer. (You may use any result proven in class.)
Show your working below and on the next page.
(a) (i) g(101011) = 4− 2 = 2.
g(100100) = 2− 4 = −2.
(ii) No. For (110), (101) ∈ S, we have g(110) = g(101) = 2− 1 = 1.
(iii) Yes. For positive integer n, take the string s = 11 . . . 1 ∈ S (repeated n times).
Then g(s) = n− 0 = n.
For n = 0, take the string s = 10 ∈ S. Then g(s) = 1− 1 = 0.
For negative integer n, take the string s = 10 . . . 0 ∈ S (with 1− n 0’s). Then
g(s) = 1− (1− n) = n.
(b) Yes. h is a bijection.
Surjective: We use the result that
Every positive integer m can be written as a sum of distinct powers of 2.
So m = an × 2n + an−1 × 2n−1 + · · ·+ a1 × 2 + a0 for some n ≥ 0 where ai = 0 or 1
with an = 1.
Then s = anan−1 . . . a1a0 is an element in S and h(s) = m.
So h is surjective.
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PAGE 15 MA1100
(More working spaces for Question 7)
(b) (cont.)
Injective: This is the same as showing, for any m ∈ Z+, we can only write m as the
sum of distincts power of 2 in exactly one way.
Proof by strong PMI:
Let P (m): There is only one way to express m as a sum of distinct powers of 2.
Basis step P (1):
Note that for n ≥ 1, the sum an × 2n + an−1 × 2n−1 + · · ·+ a1 × 2 + a0 > 1.
Therefore the only way to express 1 as sum of power of 2 is a0 = 1.
Inductive step P (1) ∧ P (2) ∧ . . . ∧ P (k)→ P (k + 1).
If k + 1 is a power of 2, say 2h, then:
(i) if n < h, then an×2n +an−1×2n−1 + · · ·+a1×2 +a0 < 2h, (and hence 6= k+ 1);
(ii) if n > h, then an×2n +an−1×2n−1 + · · ·+a1×2+a0 > 2h , (and hence 6= k+1);
(iii) if n = h, then an × 2n + an−1 × 2n−1 + · · ·+ a1 × 2 + a0 > 2h if some ai 6= 0 for
some i < n , (and hence 6= k + 1).
Hence the only way to express k + 1 = 2h as a distinct powers of 2 is 2h itself.
If k+1 is not a power of 2, let 2h be the largest power of 2 smaller than k+1. Then
k + 1 = 2h + m with m < 2h < k + 1.
By induction hypothesis, P (m) is true. So there is a unique way to write m as sum
of power of 2, say
m = 2t + at−1 × 2t−1 + · · ·+ a1 × 2 + a0
with t < h.
Since 2h is the largest power of 2 smaller than k + 1, so 2h is unique.
Hence k + 1 can be uniquely written as 2h + 2t + at−1 × 2t−1 + · · ·+ a1 × 2 + a0.
By strong PMI, we have proven that, for all m ≥ 1, m can be written as the sum
of distincts power of 2 in exactly one way.
We can then conclude that the function h is injective.
Continue on page 18-19 if you need more space. Please indicate clearly.
· · · − 16−
PAGE 16 MA1100
Question 8 [10 marks]
Let pn be the nth prime number, i.e. p1 = 2, p2 = 3, p3 = 5 . . . etc.
(a) Is it true that p1p2 · · · pn ≡ 2 mod 4 for all n ∈ Z+? Justify your answer.
(b) Prove that pn < 22n for all n ∈ Z+.
Show your working below and on the next page.
(a) Yes. p1 = 2 and pi is odd for all i ≥ 2.
So p1p2 · · · pn is even and p2 · · · pn is odd for all n ∈ Z+.
Hence p1p2 · · · pn ≡ 0 or 2 mod 4 for all n ∈ Z+.
Suppose p1p2 · · · pn ≡ 0 mod 4 for some n. Then p1p2 · · · pn = 4k for some integer
k and p2 · · · pn = 2k which contradicts that p2 · · · pn is odd.
So we conclude that p1p2 · · · pn ≡ 2 mod 4.
(b) We shall prove by strong mathematical induction.
Let P (n) : pn < 22n .
Basis step: Let n = 1. We have p1 = 2 < 221 . So P (1) is true.
Inductive step: Assume P (1), P (2), . . . , P (k) are true. i.e. pi < 22i for all i =
1, 2, . . . , k. We can also write this inequality as pi + 1 ≤ 22i .
Let N = p1p2 · · · pk + 1. Then by hypothesis,
N = p1p2 · · · pk + 1
< (p1 + 1)(p2 + 1) · · · (pk + 1)
≤ (221)(222) · · · (22k)
= 2(21+22+···+2k)
= 2(2k+1−2)
< 22k+1
Let p be a prime factor of N . Then p 6= pi for 1 ≤ i ≤ k. Otherwise, p | N and
p | (N − 1). This implies p | 1 which is impossible.
So p is a prime bigger than pk. i.e. pk+1 ≤ p ≤ N < 22k+1.
This proves that if P (1), P (2), . . . , P (k) are true, then P (k + 1) is true.
Hence by strong mathematical induction, we have proven that pn < 22n for all
positive integers n.
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PAGE 17 MA1100
(More working spaces for Question 8)
Continue on page 18-19 if you need more space. Please indicate clearly.
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PAGE 18 MA1100
(Additional working spaces for ALL questions - indicate your question numbers clearly.)
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PAGE 19 MA1100
(Additional working spaces for ALL questions - indicate your question numbers clearly.)
[END OF PAPER]