UNIVERSITY OF GHANA (All rights reserved)
BACHELOR OF SCIENCE IN ENGINEERING FIRST SEMESTER EXAMINATIONS, 2011/2012
CENG 315/CPEN 205 DISCRETE MATHEMATICAL STRUCTURES INSTRUCTIONS:
ANSWER ALL QUESTIONS
EACH QUESTION CARRIES 25 MARKS TIME ALLOWED: TWO (2) HOURS
Q1. a) Let T(x,y) stand for “x has taken y,” where the domain of discourse for x consists of students and the domain of discourse of y consists of math courses at your school. Translate each of the following propositions into unambiguous English sentences.
i. "x"y T(x,y) [1 mark] ii. "x$y T(x,y) [1 mark] iii. "y $x T(x,y) [1 mark] iv. $x$y T(x,y) [1 mark] v. $x"y T(x,y) [1 mark] vi. $y "x T(x,y) [1 mark] b) Let L(x,y) be the statement “x loves y,” where the domain for both x and y consists of all people in the world. Use quantifiers to express each of these statements.
i. Everybody loves Jerry. [1 mark] ii. Everybody loves somebody. [1 mark] iii. There is somebody whom everybody loves. [1 mark] iv. Nobody loves everybody. [1 mark]
v. There is somebody whom Lydia does not love. [1 mark] c) Write each of these statements in the form “if p, then q” in English. [Hint: Refer to the list of common ways to express conditional statements.]
i. It snows whenever the wind blows from the northeast. [1 mark] ii. The apple trees will bloom if it stays warm for a week. [1 mark] iii. That the Pistons win the championship implies that they beat the Lakers. [1 mark] iv. It is necessary to walk 8 miles to get to the top of Long’s Peak. [1 mark]
d) Show that each of these conditional statements is a tautology by using truth tables. i. [¬p ˄ (p ˅ q)] → q [2 marks] ii. [(p → q) ˄ (q → r)] → (p → r) [2 marks] iii. [p ˄ (p → q)] → q [2 marks] iv. [( p ˅ q ) ˄ ( p → r ) ˄ ( q → r )] → r [2 marks]
Q2. a) How many license plates can be made using either two or three letters followed by either two or three digits? [5 marks] b) A particular brand of shirt comes in 12 colors, and has a male and female version. It also comes in three sizes; Medium (M), Large (L), and Extra Large (XL) for each gender. Using a tree diagram, determine how many types of this shirt are made. [5 marks] c) In the lottery of a certain country, the numbers drawn range from 1 to 49 inclusive. Bamidele, a very rich Nigerian immigrant, has become so addicted to the lottery that he stakes it both on Wednesdays and Saturdays, which are the draw days. To win the jackpot in this lottery, one has to correctly predict all the six numbers drawn (i.e., for example, 2-45-7-47-25-15 on one ticket). Anything less than six numbers but between three and five correct predictions still qualifies one to win some money on the lottery but not the jackpot. On one Saturday, Bamidele decided to go for broke and staked every possible numbers that were definitely to be drawn. This means Bamidele will surely be a jackpot winner as well as a winner of any of the available prizes as well that Saturday. Each ticket in the lottery, however, costs £1 to stake. On this particular Saturday, the jackpot which was worth £70,000,000 was won by five people including Bamidele and it was to be shared equally among them. For correctly predicting 5 numbers that very
Saturday, one also wins £100,000, for 4 correct predictions one wins £5,000 and for 3 correction predictions, one wins a token of £10. Based on the preamble given above, determine
Q3. a) i) Define a set. [1 mark] ii) Define the cardinality of a set. [1 mark] b) Compute the cardinality of each of these sets.
i. |{1, -13, 4, -13, 1}| [2 marks] ii. |{3, {1,2,3,4}, Æ}| [2 marks] iii. |{}| [1 mark] iv. |{ {}, {{}}, {{{}}} }| [2 marks]
c) A and B are subsets of a universal set U and A and B are also not disjoint. Represent them by means of a Venn diagram. Shade the regions which represent the sets.
i. A Ç B' [1 mark] ii. B Ç A' [1 mark] iii. (A Ç B') È (A' Ç B). [1 mark] iv. Hence show that (A Ç B') È (A' Ç B) = (A È B) Ç (A Ç B)'. [1 mark] d) Let A be the set students who live within one mile of school and let B be the set of students who walk to classes. Describe the students in each of these sets.
i. A Ç B [2 marks] ii. A È B [2 marks] iii. A – B [2 marks] iv. B – A [2 marks]
e) ii) A drawer contains a dozen brown socks and a dozen black socks, all unmatched. A man takes socks out at random in the dark.
1) How many socks must he take out to be sure that he has at least two socks of the same color? [2 marks] 2) How many socks must he take out to be sure that he has at least two black socks?
[2 marks]
Q4. a) i) Define a relation and hence a function. [2 marks] ii) State the vertical line test for functions. [1 mark] b) i) Let f be the function from {a, b, c, d} to {1, 2, 3, 4} with f(a) = 4, f(b) = 2, f(c) = 1, and f(d) = 3. Prove that f is a bijection. [3 marks]
c) i) Conjecture a formula for + + . . . +
i. The leaves on a tree [1 mark] ii. The stars in the sky [1 mark] iii. A chapter in a book [1 mark] iv. Ideas [1 mark]
v. 60 minutes [1 mark]
