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Sma 2100 Discrete Mathematics Question Paper
Sma 2100 Discrete Mathematics
Course:Bachelor Of Science In Computer Science
Institution: Dedan Kimathi University Of Technology question papers
Exam Year:2011
KIMATHI UNIVERSITY COLLEGE OF TECHNOLOGY
UNIVERSITY EXAMINATIONS 2011/2012
BACHELOR OF SCIENCE IN COMPUTER SCIENCE
AND BACHELOR OF SCIENCE IN ACTURIAL SCIENCE
SMA 2100 DISCRETE MATHEMATICS
DECEMBER 2011 _____ TIME
2HOURS
Instruction: Answer Question One and Any Other Two Questions
Question One (30 Marks)
a) Let f and g : be functions defined on by f (x) = and
g (x) = . Evaluate
i. f o g (-3) (2 marks)
ii. g o f (-13) (2 marks)
iii. Verify whether f is one to one function (2 marks)
b) Use truth tables to test the validity of the following argument
(5 marks)
c) Negate the following statements:
i. All children born to Actuaries are artists or doctors (2 marks)
ii. If a client defaults on loan repayment then his guarantor repay it (2 marks)
d) List down the different number systems in order of size (2 marks)
e) Use Venn diagram test the truth value of the following argument:
All scientists are mathematicians. All mathematicians are footballers. Therefore all
footballers are scientists. (3 marks)
f) Use mathematical induction to show that 4n-1 is divisible by 3 n (5 marks)
Given that A = {mouse, printer, keyboard, {monitor, CPU}} and B= {monitor, printer} and C=
{mouse, keyboard}. Find
i. A B (1 marks)
ii. (B C) (A C) (2 marks)
iii. |A| (1 marks)
iv. Is B A or B A. (1 marks)
Question Two (20 Marks)
a) Use truth tables to test if the following is a tautology or a contradiction.
~(P Ú (Q ? R))?((P?Q) Ù R) (10
marks)
b) A discrete mathematics student convinces her classmates that the following argument
valid. Test its validity.
If the software crushed, then either the programmer wrote the code wrongly or a virus
affected the software. The software crushed if and only if a virus affected the
software. The programmer did not write the code wrongly or the software did not
crush. Therefore if a virus affected the software then either the software crushed or
the programmer wrote the code wrongly (10 marks)
Question Three (20 Marks)
a) Use mathematical induction to proof that (10 marks)
b) Proof that the product of any two rational numbers is rational. (4 marks)
c) Use proof by contradiction to show that is irrational. (6 marks)
Question Four (20 Marks)
a) State without proof the pigeon hole principle. Give an example (4 marks)
b) Evaluate |A ? B| given that A={3,4,-1} and B={3,-6,4,0} (3 marks)
c) State the demorgans law of sets. (3 marks)
d) A study conducted by Wajuaji consultants of the characteristics if 200 Jua kali
artisans which had failed in business revealed that 190 of them were either
undercapitalized or had inexperienced management or had poor location. The number
with all three of the characteristics was 8. It was found that 20 artisans had
inexperienced management and poor location. The number of artisans that were
undercapitalized but had experienced management and good location was 80. Further,
30 artisans had inexperienced management but had sufficient capitalization and good
location. 14 artisans were undercapitalized and had inexperienced management.
Finally, 18 artisans were found to be undercapitalized and had poor location.
i. Illustrate the above information in a Venn diagram (8 marks)
ii. Find how many had at least one of the problems listed (1
mark)
iii. Find how many had two of the problem listed (1
mark)
Question Five (20 Marks)
a) Define the following terms giving an example in each.
i. A function (3 marks)
ii. Surjective mapping (3 marks)
b) Let A = {1,2,3,4,5} and B ={2,3,4,5,6}.
Let f : A?B be defined by f (a) = a+1 for all aÎA. Let g : B? C be defined by
g(b) =
î í ì
- ³
<
1 5
, 5
b b
b b
. Compute
i. gof (3 marks)
ii. fog (3 marks)
iii. fogofog (-2) (2 marks)
iv. gofog (11) (2 marks)
c) Let f : N ? N be defined by f (n) = 3n+4. Find f-1 (4
marks)
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