The portfolio for the Discrete Mathematics part
The assessment for this part will take the form of a portfolio containing your solutions to attached exercises.
Assessment details
All exercises in the portfolio will be marked (maximum 100 marks for the portfolio).
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Information to the submission
Please write your solutions coherently and use mathematical notation correctly, e.g. do not forget to use {}, when a result is a set.
It is very important to show your workings, not just the result. You can always get partial marks even if your final result is not correct.
You may get 0 for an exercise if only a result is given without giving a reasonable explanation. You always need to answer a question “Why?”. It is not simple, but very useful for arguing 🙂 This is completely different approach to many other coursework.
Please number the pages to help the marker.
If you write your solutions by hand, please write legibly. If you can’t find a good editor for writing math notation and would like to use the best and the most interesting one, then LaTeX is recommended, see http://en.wikibooks.org/wiki/LaTeX/Introduction, http://miktex.org/ or online version https://www.writelatex.com. It may be a good idea to learn more about this option right now!
Sources of help
The main source of guidance for solving the exercises is at the tutorials. Please also feel free to discuss the exercises with other students, but the work you submit must be the result of your own effort. Copying the work of your colleagues is not acceptable! It is fairly easy to spot when a student has done this. Lecturers reserve a right to ask you for an explanation of your work before you are awarded any marks.
Any questions and problems (e.g. if you think a question is ambiguous) can be dealt with by email. If any general issues arise, I will use time in lectures and/or Moodle to point these out.
The portfolio is an individual assignment, and so the work you submit for assessment must be your own. Any attempt to pass off someone else’s work as your own is plagiarism, which is a serious academic offence. Any suspected cases of plagiarism will be dealt with in accordance with the University regulations. All assessments (if possible) should be anonymous including all coursework. Therefore you must not identify your work using your name. Please, use only your student ID (make sure you write the correct one) to identify your work.
Exercise (Question 1 to 20 is 4 marks each, Question 21 to 24 is 5 marks each , total = 100 marks)
1. In each of the following statements replace ‘?’ by the element such that the statement is true. If there is no solution, give a reason. If there are more solutions, give all of
them. (i) {{?,?}}⊆{1,{2,3},{5,6,7}}
(ii) {?,?}∈{1,2,3,{4,5}}
(iii) {?,?}⊂{A,B,C,{D,E}}
2. Determine if the set A is finite or infinite?
Justify your answer if it is infinite; enumerate and state the number of elements of A if it is finite.
A = {k |k∈Z and k2 ≤ 25}
3. Rewrite the set by listing its elements in each of the following sets (show your working): (i) {3x-2y | x ∈{3,4,5} and y ∈{-2,-1,0,1,2}}
(ii) {n | n ∈N and n3 + 2n + 5 is a multiple of 7 and 1 < n ≤ 4}
4. Justify your answers for the following :
(i) How many subsets are there in the set {a,b,c,d} ?
(ii) How many subsets of the set {a,b,c,d} doesn’t contain the element ‘a’ ? (iii) How many subsets of the set {a,b,c,d} contains the elements ‘a’ or ‘b’ (including the subsets which contain both elements ’a’, ‘b’ )?
5. (i) How many elements are in the set {(a,b) | a,b ∈ N and 1 ≤ a ≤ b ≤ 5}. Justify your answer. (ii) Can you generalise your approach from part (i) to find how many elements are in the set {(a,b) | a,b ∈N and 1 ≤ a ≤ b ≤ n} for n ≥ 1, n ∈N.
6. Let A, B, and C be sets. Is it true that always A - (B - C) = (A - B) - C ? Either prove the statement or find a counterexample.
7. There is a group of 161 students of which 20 are taking operation systems, databases, and discrete mathematics; 36 are taking operation systems and databases; 28 are taking operation systems and discrete mathematics; 31 are taking discrete mathematics and databases; 65 are taking operation systems; 75 are taking databases; and 63 are taking discrete mathematics.
(i) Draw a Venn diagram to illustrate the information above
(ii) How many students are taking discrete mathematics or operation systems (or both) but
not databases?
(iii) How many students are taking none of the three subjects?
8. Let A = {1,4,5,8}, B = {1,4,9}, and C = {1,6,7,8}. What are the elements in the following sets? (i) (A∩C)×(A∩B)
(ii) (A∩B) U (B∩C)
(iii) Can you find a set W , |W | = 5 and define a bijective function between W and P(W )? If such a set doesn’t exist, give a reason.
9. Determine whether the function: h : N→N; h(x) = 2x is partial/total, injective, surjective, or bijective. Justify your answer.
10. Define an inverse function to the following functions (i), (ii); i f such a function doesn’t exist, give a reason. Please indicate your domain and codomain properly.
i. f : Q→Q, f(x) = 2x – 7
ii. f : Z→Z, f(x) = 2x2
11. Let f : Q→Q be defined as f(x) = 1�� for every x ∈N and x ≠ 0. Calculate the following and write
down either value or a function. Show your working.
i. f(f(f(5))) =
ii. (f ◦f)(x+1) = f(f(x+1)) =
iii. Compose the function f n-times recurrently (f ◦f ◦f···f ◦f) (x)
�� n times ��
12. (i) Give an example of a function f : N → N which is total and injective, but not surjective. If such a function doesn’t exist, give a reason.
(ii) Give an example of a function f : N→N which is partial, injective, and surjective. If such a function doesn’t exist, give a reason.
13. Determine whether the following pairs of statements are logically equivalent or not. Give a reason.
(i) p → (q → r) and (p → q) → r (ii) p → q and p ↔ q
14. Determine whether the following statement is a tautology, a contingency or a contradiction. Give a reason for your answer: (p → q)∧(¬p → p) → q
15. Let P be the proposition “there are enough vaccines” and Q be the proposition “the covid cases are increasing”. Express each of the following propositions as logical expressions: (i) If there are not enough vaccines , then the covid cases are increasing. (ii) There are enough vaccines , but the covid cases are increasing.
(iii) Either there is not enough vaccine or the covid cases are not increasing but not both.
16. Verify each of the following equivalences by writing an equivalence proof. That is, start on one side and use known equivalences to get to the other side.
(i) p → (q∧r) ≡ (p → q)∧(p → r)
(ii) (p → q)∧(p∨q) ≡ q
17. Write each of the following statements in symbolic form and determine whether they are logically equivalent. Include a truth table and a few words of explanation.
“If the number of new cases decreases, you can go back office to work ” “The number of new cases decreases, and you can go back office to work”
18. Write a negation (formal or informal) for each of the following statements. Be careful to avoid negations that are ambiguous.
(i) All airports are closed.
(ii) Some languages are difficult to learn.
(iii) All students have submitted their project online (iv) All guests come and all the guests are relatives.
19. Give a direct proof of the fact that a2−7a+10 is even for any integer a. Please note that merely giving an example will not attract any mark.
20. Write down the contrapositive and the negation of the following implication.
(i) If a and b are integers, then ab is an integer. (ii) If x2 + x−2 < 0, then x > −2 and x < 1.
21. Disprove the statement by giving a counterexample or give a proof if the statement is true: (i) ∀a ∈Z ∀b ∈Z, if |a| > |b| then ab > b2.
(ii) ∀a ∈Z ∀b ∈Z, (a2 + b2) = (a + b)2
(iii) ∀a ∈Z ∀b ∈Z, if a(a-b) = 0 then a=b.
(iv) ∀a ∈Z ∃b ∈Z, if a is a multiple of 2, b is a multiple of 3, then ab is not a multiple of 6 . (v) ∀a ∈N ∃b ∈N, if a is odd, b is odd, then (a+b) is odd
22. Rewrite the definition of a surjective function f : A → B using ∀and ∃. Write down the negation of that definition explaining when the function is not surjective.
23. Prove that for all integers n, n ≥ 1;
24. Let s(x) denote the statement “x is working from home”, h(x) denotes the statement “x is attending a meeting”. g(x) denotes the statement “x has a zoom account”. Formalise each of the following statements.
The domain for all variables is the set of all staff in a software development company.
(i) “All staff working from home has a zoom account”
(ii) “Not every staff attending meeting has a zoom account”
(iii) “There are some staff having a zoom account did not work from home” (iv) “Everyone attending meeting should have a zoom account and work from home” (v) “All staff who have zoom account must attend meetings”
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