Notes on Discrete Mathematics

... respectively Then prove that the functions f and g ◦ f are one-one but g is not one-one Solution: By definiton, it is clear that f is indeed one-one and g is not one-one But g ◦ f (x) = g(f (x)) = g(2x) ... collection of n + balls that contains at least one green ball From this collection, pick a collection of n balls that contains at least one green ball Then by the induction hypothesis, this...

Ngày tải lên: 31/03/2014, 12:20

... that overall subject (Thus, if the Document is in part a textbook of mathematics, a Secondary Section may not explain any mathematics. ) The relationship could be a matter of historical connection ... 89 89 92 93 93 95 97 97 Preface These notes started in the Spring of 2004, but contain material that I have used in previous years I would ... is prime.) else output (Not prime n smallest...

Ngày tải lên: 08/03/2014, 12:20

... ỉ é ỉì ỉ é ìỉ ểề ễ ềềí è ì í ệí é ỉì ỉ ẹểề í ểệệể ệểẹ ẹ ểệ ệá ề ắ P P P P Alice éệ P P Bob ệ ẵẳ ể ỉể ỉ P P P Carl P Diane ìỉệ ỉ ề ễ ềề ì ỉể é ệề é í ểề ì ỉ ìểẹ ẹểệ è ẹểệ ì ĩ ỉéí ề ễ ềề ì ìỉệ ... S Y N b S b S Y N c S Y abc Y c S N Y ab c S N ac ệ ẵ N Y a ì ểề ỉệ c S N bc Y b ểệ ì é ỉ ề N c - ì ì ỉ ể ểể ề ỉ ỉ ì é ìá ể ì ệ ỉ ỉ ỉ ềẹ ệ ể ì ì ỉì ì ễể ệ ể ắ ỉ ì ỉ ì ề é ẹ ềỉìá ỉ ệ ìéỉ ì ắề

Ngày tải lên: 12/05/2014, 02:28

... Logic • Bit operations • 10 1 010 011 is a bit string of length nine 1. 19 Logics Tautology and Contradiction Tran Vinh Tan Definition A compound proposition that is always true (false) is called ... false Contents Propositional Logic Definition (Penguin Encyclopedia) The formal systematic study of the principles of valid inference and correct reasoning Definition (Discrete Mathematics...

Ngày tải lên: 31/05/2016, 01:24

... IMF Occassional Paper, No.22, 19 83 McKinnon, R The Order of Economic Liberalization: Financial Control in the Transition to a Market Economy, 2nd edition, The Johns Hopkins Studies in Economic Development, ... before moving on to the discussion of the reasons as to why financial liberalization might have led to banking crises.6 It is known that the early record of financial libe...

Ngày tải lên: 24/10/2012, 09:11

... TCBinh_Laptop_Y0809/_COURSEs_Y0809S1/1_Electrica LECTURE NOTES ON ELECTRICAL & ELECTRONIC TECHNIQUE By Nguyen Nhu Phong Electric Circuits Electric Machines Electronic devices & systems Small Signal Amplifiers ... (I t ), N = const , R f = const Iđm GVC ThS Nguyễn Như Phong It ↓ ↓ → E ↑ → IK → φ It m ↓ → Ut ↓ 1999 KTHTCN - ĐHBK Máy Điện 28/35 KTĐ & ĐT MPMC KÍCH TỪ HỖ...

Ngày tải lên: 25/10/2013, 13:15

... Registration Phone: 6 2-2 1-8 3 1-7 304 Fax: 6 2-2 1-8 3 1-7 306 JAPAN Prometric KK Kayabacho Tower 15F 1-2 1-1 Shinkawa Chuo-ku Tokyo 10 4-0 033 Japan Registration Phone: 8 1-3 -5 54 1-4 800 Fax: 8 1-3 -5 54 1-4 810 KOREA The ... immediately TOEFL ® Test- Prep Materials from the ETS ® THE Educational Testing Service® offers several TOEFL exam study gu...

Ngày tải lên: 01/11/2013, 15:20

... The correct NVP setting is necessary; it has an influence on the determination of the cable length by the instrument Notes on the use of hand-held test devices for installation of the KRONE Cat.6 ... The correct NVP setting is necessary; it has an influence on the determination of the cable length by the instrument Notes on the use...

Ngày tải lên: 21/12/2013, 18:15

... right in doing experiments To keep the emphasis on the discrete mathematics, logic, and computability, we’ll introduce new Prolog tools in the experiments where they are needed 1.1 Getting Started ... grandmother, grandfather, grandson, and granddaughter b Define and test the relations aunt, uncle, niece, nephew, and the maternal and fraternal versions of grandmothe...

Ngày tải lên: 21/02/2014, 09:20

... WALLS 49 PERSONAL CLEANLINESS 52 CHATTERING HOPES AND ADVICES 54 OBSERVATION OF THE SICK 59 CONCLUSION 71 APPENDIX 77 NOTES ON NURSING: WHAT IT IS, AND WHAT IT IS NOT Notes on Nursing [Sidenote: ... no one practice against which I can speak more strongly from actual personal experience, wide and long, of its effects during sickness observed Notes on Nursing 39 both upon other...

Ngày tải lên: 06/03/2014, 20:20

... functions and in growth analysis Some brief remarks on the income -education- ability interrelation conclude the comment I THE ROLE OF EDUCATION IN AGGREGATE PRODUCTION FUNCTIONS AND IN GROWTH ... through the latter parts of this paper as the discussion turns to the implications of the ability -education- income inter- relationships for the asse...

Ngày tải lên: 06/03/2014, 21:20

... mentioned monograph Studies on Predication in Russian, II: On the Predicative Use of the Russian Infinitive (section 4.2), available from the RAND Corporation Further automatic recognition routines ... Kuno and Oettinger18) The first step in a semantic analysis of Russian predicative infinitives would presumably imply an automatic separation of predicative and non -pr...

Ngày tải lên: 07/03/2014, 18:20

... k(k + 2) 1 − k k+2 for any positive integer k, so 1 1 1 + + + + + 15 35 63 99 14 3 1 1 1 + + + + + = × 3 × 5 × 7 × 9 × 11 11 × 13 1 1 1 = − + − + ··· + − 3 11 13 1 = × 1 = 13 13 Example 10 If ab ... 3x 5x − (1 + 3x) 15 − 2x + 16 − 2x =1 = = , 1 15 15 15 10 − 6x x 14 x − (10 − 6x) 10 − 13 x x 2x − − = − = , 2 14 14 x− it follows that 16 − 2x 10 − 13...

Ngày tải lên: 11/03/2014, 16:55

... around the rural districts of Hanoi, which is out of the scope of this study Future field collection, both in the rural and urban parts, may help to re-confirm the occurrence of these species in Hanoi ... The following list of sampling localities contains the collecting codes used also in the results and discussion section (a) Still waters: - Ponds, lakes: H...

Ngày tải lên: 14/03/2014, 10:20