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Sma2214:Dicrete Mathematics Question Paper
Sma2214:Dicrete Mathematics
Course:Bachelor Of Business Information Technology
Institution: Meru University Of Science And Technology question papers
Exam Year:2011
QUESTION ONE - (30 MARKS)
a) Given the sets U, A, B and C where ?? = ??:?? ? ??,-3 = ?? = 6 ,?? = 2,3,4 , ?? = -2,-1,3,4,5 , ?? = -2,3,5 , List the members of the following sets:
i. ?? (1 Mark)
ii. (?? ? ??)?? n ?? (3 Marks)
iii. ????? (2 marks)
b) Consider the following propositions: P: Monday is the first day of the month q: 42 = 16 r: Base angles of an isosceles triangle are equal
using these propositions, find the truth values of the following: i. (?????) ? ¬?? (2 Marks)
ii. ¬?? ? ?? ? ?? (2 Marks)
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iii. ?? ? ?? ? ?? (1 Marks) c) Find all integers ?? such that 1 < 4?? - 12 < 28 (3 Marks)
d) Given that ?? ?? = ?? + 4 and ?? ?? = ??2 - 16, compute: i. ?? ?? ?? ,?????????? ?? ?? 2 (2 Marks)
ii. ?? ° ?? (??) (3 Marks)
e) Show that ?? ? ?? ?? = ???? n ???? (5 Marks)
f) Prove by contradiction that ??2 + 3?? + 2 is even. (5 Marks)
QUESTION TWO – (20 MARKS)
a) In a group of 30 people, 8 speak English, 12 speak Spanish and 10 speak French. It is known that 5 speak English and Spanish, 5 Spanish and French, 7 speak English and French and 14 speak both three languages. Find: i. The number that does not speak any of these languages. (5 Marks) ii. The number that speaks French only. (3 Marks)
b) Given the sets ?= 3,-4, 7 8,31 2
0,
-4 7
, 3 3 ,??,?? , list the members of the following
sets; i. Natural numbers (2 Marks) ii. Integers (2 Marks) iii. Rational numbers (2 Marks) iv. Whole numbers (2 Marks)
c) Solve for x in -2 < 2?? + 1 = 3 (2 Marks)
QUESTION THREE – (20 MARKS)
a) Determine whether the following proposition is a contradiction using a truth table. (10 Marks) ?? ? ?? ? ?? = ?? ? ¬?? ? ??
b) Given ?? ?? = 2??_3 3??-5
and ?? ?? = 3?? + 5; both defined on the set of Real numbers,
compute:
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i. ??-1(??) (3 Marks)
ii. ???? 2 (4 Marks) iii. ?? ?? + 1 ??(??) (3 Marks)
QUESTION FOUR – (20 MARKS)
a) Use mathematical induction to prove the proposition that the sum of the first n positive integers is 1 + 2 + 3 + ?+ ?? = ??(??+1) 2 (9 Marks)
b) Let ??(??) be the statement "1 = ?? = 6". What are the truth values of: i. ?? 1 ii. ??(4) iii. ??(6) (3 Marks)
c) (6 Marks)
d) Identify the pairs of statements as part (c) that are logically equivalent. (2 Marks)
QUESTION FIVE – (20 MARKS)
a) Consider the following propositions: a: I will eat my socks b: it is snowing c: it will go jogging
Represent the sentences below using logical operators (connectives) i. Whether or not it is snowing, I will go jogging (2 Marks) ii. I will neither eat my socks nor go jogging (2 Marks) iii. If it is snowing, I will eat my socks, but I will not go jogging. (2 Marks)
b) Write the negation of each of these statements (3 Marks) i. 2 + 1 = 3 ii. Today is Thursday iii. The weather is hot and sunny
c) List the members of i. ?? × ?? ii. ??(??) where ?? = ??:1 = ?? = 4 ?????? ?? = {??,??,??} (5 Marks)
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d) Given that the function ?? ?? = ???? + ?? maps 2,3 ?????? 4 into 5,7 ?????? 9 respectively, find the values of ??,?? and hence the function ??(??) (6 Marks)
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