Analysis 1 (MATS2101)
Mathematics and Computer Science - MCS
Semester: First Semester
Level: 200
Year: 2014
FIRST SEMESTER EXAMINATION
DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE
CO U RS E
TIT LE: AN ALYS IS I
SE M EST ER : 1 A CAD EMI C Y EAR : 2 013 -2 0 14
COURSE CODE: MAT S2101 Time allowed: 3 Hours
Answer all questions. All necessary work must be shown.
1. What does it mean to say that:
i) a set A is a subset of another set B
ii) two sets A and B are disjoint
iii) a relation ℛ ⊂ A x B is a partial order
iv) a relation ℛ ⊂ A x B is a partial order
v) a function f: A → B is injective
vi) a number x is a real number
vii) a real number x is an upper bound for a set A of real numbers
viii) a real number x is the sepremum of a set A of real numbers
ix) a real number x is the maximum of a set A of real numbers
x) a series
∑
of real numbers converges
(2 points each)
For questions 2 to 6, write down only the letter of the correct answer:
2. Which of the following are representatives of the real number [{0}]?
a)
n≥1 b)
n ≥1 c)
n ≥1 d) all of the above e) only (c) 4pnts
3. Which if the functions are bijective? a) f : (-1 1) → (-1 1), x ↦ f(x) = x
3
b) f : (0 1) → (0 1), x ↦f (x) = x
3
c) ) f : [-1 1] → ℝ, x ↦ f(x) = x
3
d) f : [0 1] → [0 ∞), x ↦f (x) = x
3
4pnts
4. Which of the following inequalities are true for arbitrary real numbers x and y?
a) || x|| + |y|| ≥ |x +y|, b) | x+ y| ≤ |x| +|y|, c) |x| < |-x|, d) ||x| - |x + y| 4pnts
5. Which of the following inclusions are true?
a) ℚ ∪ ℝ ⊂ ℝ b) ℝ ∩ ℤ = ℤ c) ℝ ∪ ℤ = ℤ d) ℤ ∖ ℝ = ℕ e) none of the above
4pnts
6) Which of the following statements are true?
a) there is real numbers between every two natural numbers b) natural numbers are not dense
in themselves c) bounded sets of real numbers always have a minimum d) Every set of real
numbers contains an integer 4pnts
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7) Prove that the limit of a convergent sequence of real numbers is unique 5pnts
8) Prove that the canonical Embedding E: ℚ → ℝ, x ↦ E(x) =[{x}] is injective 5 pnts
9) Prove that the following statements about the set A ={x ∈ ℝ : 0 < x ≤ 1}:
i) the infinum of A is 0. 5pnts
ii)
is not an upper bound for A. 2 pnts
iii) the minimum of A is 1. 3 pnts
10) Using mathematical induction, show that for all x, y, z ∈ ℕ, x + (x + y) = (x + y) + z 6 pnts
11) Prove that if {x
n
} and {y
n
} are convergent sequences of real numbers then so is the
sequence {x
n
+ y
n
}. 5pnts
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