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(math110)[2006](f)final~PPSpider^_10425.pdf
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HKUST
MATH110 CONCEPTS IN MATHEMATICS
Final Examination Name:
20th December 2006 Student I.D.:
12:30C14:30

Directions:
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Do NOT open the exam until instructed to do so.

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Please write your name, ID number, and Section in the space provided above.

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Answer ALL questions.

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This is a closed book examination.

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You may write on both sides of the examination papers.

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You must show the working steps of your answers in order to receive full marks.

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All mobile phones and pagers should be switched o. during the examination.

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Cheating is a serious o.ense. Students who commit this o.ense may receive zero mark in the examination. However, more serious penalty may be imposed.


Question No. Points Out of
Q. 1 11
Q. 2 15
Q. 3 15
Q. 4 15
Q. 5 14
Q. 6 15
Q. 7 15
Total Points 100

. To verify your identity, you must sign your name here:

Answer each of the following questions.
1. (a) De.ne what is a function. [1]
Ans.
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(b) Let : Z . Z, be de.ned by (n) = n + 2 1 2 (n + 1) if n is even; if n is odd.
i. Determine with full justi.cation of whether is injective. [3]
Ans.
ii. Determine with full justi.cation of whether is surjective. [4]
Ans.
(c) Let f : R . [.1, 1], be given by f(x) = sin x. Find f.1 . [. 3 2 , 3 2 ] . . [3]

Ans.

2. (a) Let be a relation on a set A. Explain what is an equivalence relation on A. [2]
Ans.
(b) Let be a relation de.ned on N N in such a way that (a, b) (c, d) if and only if a + d = b + c. Show that is an equivalence relation on N N [4]
Ans.
(c) Identify the following equivalence classes. i. [(2, 5)] [2]
Ans. ii. [(3, 4)]. [2]
Ans.

(d) Identify all the equivalence classes under and describe how they partition the set N N. What is the cardinality of the set of equivalence classes? [5]
3. (a) Find two integers x and y such that

gcd(1234, 4321) = 1234x + 4321y.
[5]
Ans.

(b) Find the reminder when 15 +25 + + 995 + 1005 is divided by 4. [5]
(c) Let a, b be two integers. Prove that if k> 0, then gcd(ka, kb)= k gcd(a, b). [5]

Ans.

4. Let b> 0 and S be a set of positive real numbers which is bounded above.
(a) De.ne the least upper bound of S. [2]
Ans.
(b) Prove that the least upper bound of S in (a) is unique. [3]
Ans.
(c) i. De.ne bS = {bs : s S}. Prove that sup(bS + c)= b sup(S)+ c.
[8]
Ans.
ii. What should we modify the above formula in (c) if b< 0? [2]
5.
(a) State the Fermat little theorem. [2]


6.
Let f : X . Y be a function.



Ans.
(b) Find the reminder when 116116611 is divided by 117. [4]
Ans.
(c) Find a solution to x3 4 mod 17 and verify your solution. [8]

(a) Prove that if f is injective, then f(A) f(B)= f(A B) for any two subsets A, B of X. [7]

Ans.

(b) If f(A) f(B)= f(A B) for any two s