ATILIM UNIVERSITY DEPARTMENT OF MATHEMATICS Math 211 - Discrete Mathematics with Applications 2010-2011 Fall Semester Problem Set I Prepared by Mehmet TURAN

O−, Ω−, Θ− Notations
1. Let f and g be real valued functions deﬁned on the same set of nonnegative real numbers. (a) Prove that if g(x) is O(f (x)), then f (x) is Ω(g(x)). (b) Prove that if f (x) is O(g(x)) and c is any nonzero ral number, then cf (x) is O(cg(x)). (c) Prove that if f (x) is O(h(x)) and g(x) is O(k(x)), then f (x) + g(x) is O(G(x)), where, for each x in the domain, G(x) = max(|h(x)|, |k(x)|). (d) Prove that f (x) is Θ(f (x)). (e) Prove that if f (x) is O(h(x)) and g(x) is O(k(x)), then f (x)g(x) is O(h(x)k(x)). 2. (a) Show that for any real number x, if x > 1 then |x3 | ≤ |2x3 + 3x + 4|. (b) Show that for any real number x, if x > 1 then |2x3 + 3x + 4| ≤ 9|x3 |. (c) Use the Ω− and O− notations to express the results of parts (a) and (b). (d) What can you deduce about the order of 2x3 + 3x + 4? 3. Use the deﬁnition of Θ− notation to show that 3x5 + 7x2 + 5 is Θ(x5 ). 1 4. (a) Show that for any real number x, if x > 1 then | 2 x2 − 14x + 5| ≤ 20|x2 |.

(b) Use the O− notation to express the result of part (a). 5. (a) Show that for any real number x, if x > 1 then | 1 x5 − 3x3 + 2x − 5| ≤ 11|x5 |. 3 (b) Use the O− notation to express the result of part (a). 6. Show that x4 is not O(x2 ). 7. Use the deﬁnition of Ω− notation to show that 6x3 − 13x − 5 is Ω(x3 ). 8. Use the deﬁnition of Ω− notation to show that 9. Let n be a positive integer. (a) Show that 12 + 22 + 32 + · · · + n2 is Θ(n3 ). (b) Show that 13 + 23 + 33 + · · · + n3 is Θ(n4 ). 1 1 2 2x

− 3x − 5 is Ω(x2 ).

(c) Show that 2 + 4 + 6 + · · · + 2n is Θ(n2 ).
n

(d) Show that
i=1 n

(6i − 5) is Θ(n2 ). i(i + 3) is Θ(n3 ).
i=1 n

(e) Show that

(f) Show that
k=1

(k 2 − 2k) is Θ(n3 ).

10. Use the inequality n < 2n to show that log n is O(n).

...atCS203 Homework 6
Section 9.1 Question 17 Solution: Show that the following identities hold. a) x ⊕ y = (x + y)(xy) b) x ⊕ y = (xy) + (xy) x 0 0 1 1 y 0 1 0 1 x⊕y 0 1 1 0 (x + y) 0 1 1 1 (xy) 1 1 1 0 (x + y)(xy) 0 1 1 0 (xy) 0 0 1 0 (xy) 0 1 0 0 (xy) + (xy) 0 1 1 0
Section 9.1 Question 20 Solution: Find the a) b) c) d) duals of the following Boolean expressions. x+y → xy xy → x+y xyz + xyz → (x + y + z)(x + y + z) xz + x · 0 + x · 1 → (x + z) · (x + 1) · (x + 0)
Section 9.1 Question 23 Solution: How many diﬀerent Boolean functions F (x, y, z) are there so that F (x, y, z) = F (x, y, z) for all values of the Boolean variables x, y, and z. The solution to this problem is to realize that by inverting the bits, the actual function is inverted. So for F (x, y, z) = F (x, y, z), then the ﬁrst four lines of the function must be an inversion (or mirror image) of the last four lines. So the number of possible combinations is 24 or 16, instead of 28 . x 0 0 0 0 1 1 1 1 y 0 0 1 1 0 0 1 1 z 0 1 0 1 0 1 0 1 x 1 1 1 1 0 0 0 0 y 1 1 0 0 1 1 0 0 z 1 0 1 0 1 0 1 0 (x + z) 0 1 0 1 1 1 1 1 (x + z) 1 1 1 1 1 0 1 0
Section 9.2 Question 3 Solution: Find the sum-of-products expansions of the following Boolean functions.
d)F (x, y, z) = x + y + z = x + y(x + x) + z(x + x) = x + xy + xy + xz + xz = x(z + z) + xy(z + z) + xy(z + z) + xz + xz 1
= xz + xz + xyz + xyz + xyz + xyz + xz + xz = xz(y + y) + xz(y + y) + xyz + xyz + xyz + xyz + xz(y + y) + xz(y + y) = xyz + xyz + xyz + xyz...

... |Information and Communication Technology (KICT) |
|Department / Centre |Computer Science (CS) |
|Programme |Bachelor of Computer Science (BCS) |
|Name of Course / Mode |DiscreteMathematics |
|Course Code |CSC 1700 |
|Name (s) of Academic staff / |Assoc. Prof. Dr. Azeddine Messikh |
|Instructor(s) | |
|Rationale for the inclusion of the course|To introduce students DiscreteMathematics and its basic principles. |
|/ module in the programme | |
|Semester and Year Offered |Semester 1 and 2 |
|Status...

...Title #1: Primadona 50% and 30% sale
Description: 5 items (Shoes, Dress, T-shirt, Bags, Accessories) has 50% sales off. Input the Product Code, the Original Price, the Quantity of the Product that you will buy then the program will show the Product Description and the Discounted Price.
Product Description Product Code Original Price
Shoes S or s 1,500
Dress D or d 550
T-shirt T or t 230
Bags B or b 1,200
Accessories A or a 300
Product Layout:
Title #2: Menu
Description: Input the amount of money that you have, then the program will show the food that you can get with the amount that you enter. And if you enter less than 25 pesos the program will display “No Food is Available”.
Food Amount
Vegetable dishes 25-30 pesos
Fried chicken 40-55 pesos
Liempo 56-95 pesos
Chicken Inasal 96-150 pesos
Seafood platter 170-180 pesos
Lechon de leche / Lechon 181-above
.
Title #3: Flower’s Symbols
Description: Input the code of the flower, then the program will show the symbols of the flower that you want to know.
hsdfggggggggggggggssssssssssssssskhfkjwenkchkewndlihwqoihfui-
ewygcbjsdejchfewuhjkfqebcbeqjbcjhqe
fewkjbfjhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh-
hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh-
hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh-...

...Mathematical ideas are often divided into two types, those that are
continuous, and those that are discrete.
An example of continuous is the number line. Between any two points,
there are always more points.
For discrete sets, this is not true. For instance, in baseball there
are four bases. If you get a hit it is either a one-base hit (what we
call a single), a two-base hit, a three-base hit, or a home run.
There is no such thing as a 2 1/2 base hit.
Discrete things are found in bundles or lumps, and you can only have
certain numbers of them.
Money is another discrete idea because you can not sell anything for
$0.005. Prices can be grouped for specials, like 2 for 99 cents, but
if you buy one it is either 49 cents or 50 cents. Discrete does not
mean it has to be whole numbers, but it does mean there are only some
that can be chosen, and some can not.
Discrete sets can be infinite, but they can not be infinitely
divisible. For example, the counting numbers from 1 to infinity are
discrete, because, like the bases in baseball, you go from one to two
and then to three but not the points in between. The number line from
0 to 1 is not discrete but continuous, because between any two points
in the set, there is always another point. This is the key that makes
the difference. In discrete we can talk...

...men like Rene Descartes who said, “I think therefore I am”, and finally to the unprecedented discoveries in the fields of mathematics and science. Among all the civilizations of time, those of the Pre-Columbian Era seem to have successfully applied mathematical concepts, mainly geometry and algebra, in a somewhat uncanny manner. One cannot all but question how engineers of today’s time, men and women with almost limitless resources, suffer periodic setbacks, while structures of the primitive Pre-Columbians have remained largely intact up until the present day. Clearly no one can compare the Golden Gate Bridge, Lincoln Tunnel, and Empire State building to Pre-Columbian structures, yet the simplistic success of these ancient people causes substantial curiosity. It seems, although only a personal conjecture, that through the analysis of modern day mathematics, insight into the minds of the long lost masterminds behind some of the worlds greatest architecture and the mathematics emphasized in their extraordinary works, can be ascertained.
The ancient Maya, although a civilization that first emerged during the pre-classic period, actually have a lot of similarities to the people of the modern era. Socially, politically, and even creatively, they were far more advanced then many may have assumed. However, the advancements that the Mayans made in mathematics were both intriguing and impressive. Formally, the Mayans are...

...History of mathematics
A proof from Euclid's Elements, widely considered the most influential textbook of all time.[1]
The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past.
Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. The most ancient mathematical texts available arePlimpton 322 (Babylonian mathematics c. 1900 BC),[2] the Rhind Mathematical Papyrus (Egyptian mathematics c. 2000-1800 BC)[3] and the Moscow Mathematical Papyrus (Egyptian mathematics c. 1890 BC). All of these texts concern the so-calledPythagorean theorem, which seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry.
The study of mathematics as a subject in its own right begins in the 6th century BC with the Pythagoreans, who coined the term "mathematics" from the ancient Greekμάθημα (mathema), meaning "subject of instruction".[4]Greek mathematics greatly refined the methods (especially through the introduction of deductive reasoning andmathematical rigor in proofs) and expanded the subject matter of mathematics.[5] Chinese...

...Week 5
Final Exam
Continuous schedule from Friday , November 1st. 9am until Saturday , November 2nd., 23:59pm.
Monday, November 4, 2013
20%
100%
To obtain the opportunity to take your final exam you should have delivered at least 6 activities.
Please keep this Agenda at hand so that you can deliver you assignments on time.
Greetings,
Blanca Alanís
Posted by: BLANCA HILDA ALANIS PENA
Posted to: CEL.HI09107V.343.13320 Inglés VII
Bibliography
Posted on: Thursday, October 3, 2013
Hello guys,
The books we are going to use are:
Text book:
Richards, Jack C. & Sandy, Chuck (2009). Passages 2 (2nd ed.). New York, N.Y. Cambridge University Press.
ISBN 978-0-521-68391-3
Workbook:
Richards, Jack C. & Sandy, Chuck (2009). Passages 2 (2nd ed.). New York, N.Y. Cambridge University Press.
ISBN 978-0-521-68393-7
Make sure they are the 2nd. edition, because the 1st. edition is completely different.
In your course, in the Bibliography Section you have a link of a bookstore where you can buy the books. You can try other bookstores in your city, of course, but they don't usually have the book in stock.
Greetings,
Blanca Alanís
Posted by: BLANCA HILDA ALANIS PENA
Posted to: CEL.HI09107V.343.13320 Inglés VII
Grading in the courseWeek 5
Final Exam
Continuous schedule from Friday , November 1st. 9am until Saturday , November 2nd., 23:59pm.
Monday, November 4, 2013
20%
100%
To...