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Basic Mathematics V1

Assignment – A

1. a) What is a Set? Explain various methods to represent a set in set theory.

b) Define the following with the help of suitable examples.

Singleton Set Finite Set

Cardinality of a Set Subset of a Set.

2. a) Define logical statement. What is a truth table? Prepare the truth tables for the

following statements and then check which are the tautologies.

q pq

b) Define Conjunction Disjunction with example.

If p: He is smart

Q: He is rich

Give a simple verbal proposition for each of the following propositions

pq pq p

3. a) What are the advantages of measures of central tendency? Discuss various

measures of central tendency.

b) Find the mean, median mode for the following data.

Marks

0-10

10-20

20-30

30-40

40-50

50-60

60-70

70-80

80-90

90-100

Students

2

14

10

7

9

15

2

9

3

1

4. An art museum has arranged its current exhibition in the five rooms shown in figure 1.

Is there a way to tour the exhibit so that you pass through each door exactly once? If so,

give a sketch of your tour.

5. Modify Kruskal’s algorithm so that it will produce a maximal spanning tree, that is,

one with the largest possible some of weights.

Figure 1

Assignment – B

1. a) Describe all the relationships seen in this Venn diagram:

b) Draw the Venn diagram for AB= Ø.

c) Draw a Venn diagram to prove the third subset theorem : If A, B, C are sets with

and then .

2 a) Prove that following two statements are contradictions.

b) Prove that following are the equivalent statements.

p

3. Let G be a group. Show that the function f.G G defined by f(a)=a-1 is an

isomorphism if and only if G is Abelian.

4. Consider the labeled tree whose diagraph is shown is the following figure. Draw the

graph of the corresponding binary positional tree B(T) show their correspondence

to vertices of T

S

t u

v w x v z

Assignment – C

In question 1 through 5, classify the sets as finite or infinite set.

1. Set of four seasons in year

2. Set of vowels in word “fitted”

3. Set of multiples of 8 more than 95 and less than 97

4. Set of months of the year

5. Set of vowels in word “command”

In question 6 through 10, classify the sets as empty or singleton set.

6. Set of multiples of 2 more than 0 and less than 4

7. {0}

8. Set of vowels in word “ call”

9. Set of vowels in word “ fair”

10 {9}

In question 11 through 17, classify the non – equivalent or equal sets.

11. A= Set of vowels in word “ bottom”, B= Set of vowels in “word bottom”

12. A= {a, b, c, d, e, f, g ,h, x} B== {1 ,2, 3, 4, 5, 6, 7, 8, 24}

13. A = Set of vowels in word “March”, B = Set of vowels in word “May”

14. A = Set of multiples of 12, B = {12, 24, 36……}

15. A = Set of letters in “finance”, B = Set of letters in “mathematics”

16. A = Set of multiples of 12, B = {12, 24, 36……}

17. A = Set of multiples of 7, B = {7, 14, 21……}

In question 18 through 21, find the truth table.

18. For every integer n, (2n+1) is an even integer.

19. f(x) = Cos x implies f’(x) = -Sin x

20. The sum of one even and one odd integer is even integer

21. 5+6=11

22. Calculate the arithmetic mean of 5.7, 6.6, 7.2, 9.3, and 6.2.

In question 23 through 25, the marks obtained by 12 students in a class test are

14, 13, 09, 19, 05, 08, 16, 17, 11, 10, 12, 16.

Find

23. The mean of their marks.

24. The mean of their marks when the marks of each student are increased by 3.

25. The mean of their marks when the marks of each student are doubled.

In exercise 26 through 37, choose the correct answer:

26. The number of vertices of odd degree in a graph is

a) always even b) always odd

c) either even or odd d) always zero

27. A vertex of degree one is called as

a) Rendant b) isolated vertex

c) null vertex d) colored vertex

28. A circuit in a connected graph, which includes every vertex of the graph is known

a) Euler b) Universal

c) Hamilton d) None of these

29. A given connected graphic is a Euler graph if and only if all vertices of G are of

a) same degree b) even degree

c) odd degree d) different degrees

30. The length of Hamilton path (if exists) in a connected graph of n vertices is

a) n-1 b) n

c) n+1 d) n/2

31. A graph with n vertices and n+1 edges that is not a tree, is

a) connected b) disconnected

c) eculer d) a circuit

32. A graph is a tree if and only if

a) is completely connected b) is minimum connected

c) contains a circuit d) is planer

33. The minimum number of spanning trees in a connected graph with n nodes is

a) 1 b) 2

c) n-1 d) n/2

34. The number of different rooted labeled trees with n vertices

a) 2n-1 b) 2n

c) nn-1 d) nn

35. The number of circuit in a tree with n nodes

a) zero b) 1

b) n-1 d) n/2

36. Which of the following is false ?

a) The set of all objective functions on a finite set forms a group

under function composition

b) The set {1,2,…p-1} forms a group under function composition.

c) The set of all strings over a finite alphabet forms a group under

concatenation.

d) A subset of G is a sub group of the group (G,*) if and only if for

any pair of elements a, b D, a*b-1 S

37. Let (Z, *) be an algebraic structure where Z is the set of integers and the operation *

is defined by n * m = maximum (n, m), which of the following statements is true for

(z,*).

a) (z,*) is a monoid b) (z,*) is an Abelian group

c) (z,*) is a group d) None of the above

In question 38 through 40 determine whether the set together with binary operation is a

group.

38. Z, where * is subtraction

39. R, where a * b = a + b + 2

40. The set of all matrices under the operation of matrix addition

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