ASSIGNMENT : Basic Mathematics

PROGRAM: B.Sc. – IT

SEMESTER-I

Subject Name: Basic Mathematics

Permanent Enrollment Number (PEN) : A1922713004(el)

Student Enrollment Number (SEN) : EL2013-01-BIT0001-2402

Student Name: RANA RAM BHAJAN SINGH

Assignment - A

Q1. a) What is a Set? Explain various methods to represent a set

in set theory.

Ans 1a): A set is a group or collection of objects or numbers, considered as an entity

unto itself. Sets are usually symbolized by uppercase, italicized, boldface letters such as

A, B, S, or Z. Each object or number in a set is called a member or element of the set.

Examples include the set of all computers in the world, the set of all apples on a tree,

and the set of all irrational numbers between 0 and 1.

When the elements of a set can be listed or enumerated, it is customary to enclose the

list in curly brackets. Thus, for example, we might speak of the set (call it K) of all

natural numbers between, and including, 5 and 10 as:

K = {5, 6, 7, 8, 9, 10}

Methods to represent a set:

There are two methods to represent a set. One is Rule method, another one is Roster

method.

Rule is a method of naming a set by describing its elements.

For example, { x: x > 3, x is a whole number} describes the set with elements 4, 5, 6,….

Therefore, { x: x > 3, x is a whole number} is the same as {4,5,6,…}. { x: x > 3}

describes all numbers greater than 3. This set of numbers cannot be represented as a

list and is represented using a number line graph.

Roster is a method of naming a set by listing its members. For example, {1,2,3} is the

set consisting of only the elements 1,2, and 3. There are many ways to represent this

set using a rule.

Two correct methods are as follows:

{x: x < 4, x is a natural number} {x: 0 < x < 4, x is a whole number}

An incorrect method would be {x:0 < x < 4} because this rule includes ALL numbers

between 0 and 4, not just the numbers 1, 2, and 3.

Q1 b): Define the following with the help of suitable examples.

Singleton Set Finite Set

Cardinality of a Set Subset of a Set

Ans 1b):

Singleton Set

A set having exactly one element is called Singleton set. A singleton set is denoted by

and is the simplest example of a nonempty set. Any set other than the empty set is

therefore a nonempty set. For example {a}

Finite Sets:

Finite sets have a countable number of elements. For example, {a, b, c, d, e} is a set of

five elements, thus it is a finite set.

Cardinality of a Set:

 The cardinality of a set is a measure of the "number of elements of the set". For

example, the set A = {2, 4, 6} contains 3 elements, and therefore A has a cardinality of

3. The cardinality of a set A is usually denoted | A |, with a vertical bar on each side; this

is the same notation as absolute value and the meaning depends on context.

Alternatively, the cardinality of a set A may be denoted by n(A), or # A.

Subset of a Set:

A subset is a set contained in another set. So for example, we have the set {1, 2, 3, 4,

5}. A subset of this is {1, 2, 3}. Another subset is {3, 4} or even another, {1}. However,

{1, 6} is not a subset, since it contains an element (6) which is not in the parent set. 

Q2. 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 p q

Answer 2(a).Logic Statement:

A statement is an assertion that can be determined to be true or false.

Truth Table:

A truth table is a mathematical table used in logic—specifically in connection

with Boolean algebra, boolean functions, and propositional calculus—to compute the

functional values of logical expressions on each of their functional arguments, that is, on

each combination of values taken by their logical variables. In particular, truth tables can

be used to tell whether a propositional expression is true for all legitimate input values,

that is, logically valid.

(i),(iii),(iv)P q P

v q

P ^q

P -> q

q p q

T T T T T T T TT F T F F T T TF T T F T T T TF F F F T T T T

(ii)p Q r r v

qP^q ~(r v

q)

T T T T T F FT T F T T F FT F T T F F FT F F F F T FF T T T F F FF T F T F F FF F T T F F FF F F F F T F

1, 3 & 4 are Tautology.

Q2 b): Define Conjunction Disjunction with example.

If p: He is smartQ: He is rich

Give a simple verbal proposition for each of the following propositions

p q p q p

Solution 2b)Conjunction:Conjunction is a truth-functional connective similar to "and" in English and is represented in symbolic logic with the dot "     ". 

Disjunction:Disjunction is a connective which forms compound propositions which are false only if both statements (disjuncts) are false. It is sometimes called, alternation.

Logical conjunction

Logical conjunction is an operation on two logical values, typically the values of two propositions, that produces a value of true if both of its operands are true.

The truth table for p AND q (also written as p q∧ ) is as follows:

P Q P v q

T T TT F TF T TF F F

Logical disjunction

Logical disjunction is an operation on two logical values, typically the values of two propositions, that produces a value of true if at least one of its operands is true.

The truth table for p OR q (also written as p q∨ ) is as follows:

P q P^qT T TT F FF T FF F F

If p: He is smartQ: He is rich

Given a simple verbal proposition for each of the following propositions He is not smart

He is not rich

p q He is smart or rich p q He is smart and rich p He is smart or not rich

He is not smart or not rich

He is not smart and not rich

He is not smart or not rich

Q3 a): What are the advantages of measures of central tendency? Discuss various measures of central tendency.

Sol 3a): Advantages of measuring of central tendency:In central tendency the large group of data is grouped into a single value for effective business decision making.

Three common measures of central tendency are the mean, median and the mode. They are measures in that they tell how far any given value is from the center of a set of data. For example, the mean is commonly known as the average. The others do the same thing.

Q3 b): Find the mean, median mode for the following data.Marks 0-10 10-20 20-30 30-40 40-

5050-60 60-70 70-80 80-90 90-

100Students 2 14 10 7 9 15 2 9 3 1

MeanThe mean (or average) is the most popular and well known measure of central tendency. It can be used with both discrete and continuous data, although its use is most often with continuous data. The mean is equal to the sum of all the values in the data set divided by the number of values in the data set. So, if we have n values in a data set and they have values x1, x2, ..., xn, then the sample mean, usually denoted by   (pronounced x bar), is:

Median

The median is the middle score for a set of data that has been arranged in order of magnitude. The median is less affected by outliers and skewed data. In order to calculate the median, suppose we have the data below:

65 55 89 56 35 14 56 55 87 45 92

We first need to rearrange that data into order of magnitude (smallest first):

14 35 45 55 55 56 56 65 87 89 92

Our median mark is the middle mark - in this case 56 (highlighted in bold). It is the middle mark because there are 5 scores before it and 5 scores after it. When we have

an even number of scores, simply we have to take the middle two scores and average the result. So, if we look at the example below:

65 55 89 56 35 14 56 55 87 45

We again rearrange that data into order of magnitude (smallest first):

14 35 45 55 55 56 56 65 87 89 92

Only now we have to take the 5th and 6th score in our data set and average them to get a median of 55.5.

Mode:

The mode is the most frequent score in our data set. On a histogram it represents the highest bar in a bar

chart or histogram. Sometimes the mode is considered as being the most popular option. An example of a

mode is presented below:

b) Mean=43.06 Median=46, Mode= 53.16

Solve: Mean=∑xifi =5*2+15*14+25*10+35*7+45*9+55*15+65*2+75*9+85*3+95*1

N=72 Mean=∑xifi /N=43.06

Median=L+h/f(n/2-c)

Mode=1+h(f1-f0)/(2f1-f0-f2)

Q4: 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.

Solution: Sketch of Tour.

Q5. Modify Kruskal’s algorithm so that it will produce a maximal spanning tree, that is, one with the largest possible some of weights.

Figure 1Solution:A maximum spanning tree is a spanning tree with weight greater than or equal to the

weight of every other spanning tree.

To accomplish this with Kruskal's Algo, Perform Step 1:

Step 1:

Multiplying the edge weights by -1 of the given Graph

And taken it as new Graph and Apply Kruskal’s Algo For MST Problem.

Assignment - B

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

Sol 1 a):The red portion of this picture shows the relation of A∪B

Fig: A∪BThe red portion of the picture bellow indicate the relation of A∩B 

Fig: A∩B 

The blue portion of the picture bellow represents the relation of A′

Fig: A′

The red portion of the picture bellow represents B′

Fig: B′

Que 1 b): Draw the Venn diagram for A B= Ø.

Sol 1b):

Que 1 c): Draw a Venn diagram to prove the third subset theorem : If A, B, C are sets with

and then .

Sol 1c:

Fig: Proved of subset theorem

Q2 a): Prove that following two statements are contradictions.

Solution 2a):Contradiction: A contradictions is a formula which is "always false"; 

(i): P Q ~p ~q ~p

^ ~q

p or q

T T F F F T FT F F T F T FF T T F F T FF F T T T F FSince Last Column contains only F’s. So it is contradiction.

(ii): P q ~p ~q P ^ ~q ~p v qT T F F F T FT F F T T F FF T T F F T FF F T T F T F

Since Last Column contains only F’s. So it is contradiction.

Q2 b): Prove that following are the equivalent statements. p

Solution 2b):

Two statements X and Y are logically equivalent if   is a tautology.A tautology is a formula which is "always true".(i): p

p q r q v r

p (qvr)

p q

p r (p q)v(p r)

p

T T T T T T T T TT T F T T T F T TT F T T T F T T TT F F F F F F F TF T T T T T T T TF T F T T T T T TF F T T T T T T TF F F F T T T T TSince Last Column contains only T’s. So it is Tautology.Hence the statement is equivalent statements.

(ii): p q ~p ~q p v q ~(p v q) (~p ^

~q)T T F F T F F TT F F T T F F TF T T F T F F TF F T T F T T TSince Last Column contains only T’s. So it is Tautology.Hence the statement is equivalent statements.

Q3: 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.

Solution:Let f be an isomorphic of G.

Let a1, a2 G be arbitrary.

As given

f (a1a2) = (a1a2)-1

=> f(a1)f (a2) = a2-1

. a1-1 ; since f is a Homorphism,

=> a1-1 a2

-1 = a2-1 a1

-1, as given

=> (a1-1 a2

-1)-1 = (a2-1 a1

-1)-1

=> (a2-1)-1 (a1

-1)-1 = (a1-1)-1 (a2

-1)-1

=> a2a1 = a1a2 v a1,a2 G.

Here G is abelion.

Conversely, let G be abelion, We shall prove,

F:G→G defined by f(a)=a-1 v a G is

An isomorphsm,

Let a,b G, we have

f(ab) = (ab)-1 = b-1 a-1

= a-1b-1; since G is abelion

=> f(ab) = f(a)f(b)

Then f is an Homorphism

Now f ( a) = f (b) => a-1=b-1 => (a-1)-1=(b-1)-1

=> a=b

Then f is one –to-one.

For any a G, a= (a-1)-1 =b-1, when b= a-1 G

then a=b-1 => a=f(b) => T is Onto.

then f is an isomorphism of G.

Q4: 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

Solution 4:

z u s t

v w x

Assignment - C

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

1. Set of four seasons in year2. Set of vowels in word “fitted”3. Set of multiples of 8 more than 95 and less than 974. Set of months of the year5. Set of vowels in word “command”

Answers:1. Finite Set2. Finite Set3. Finite Set4. Finite Set5. Finite Set

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 47. {0}8. Set of vowels in word “ call”9. Set of vowels in word “ fair”10 {9}

Answers:

6. Empty Set (consist of no elements)7. Singleton Set (consist of one elements)8. Singleton Set (consist of one elements)9. Singleton Set (consist of one elements)10.Singleton Set (consist of one elements)

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......}

Answers:11.Non – equivalent Set12.Non – equivalent Set13.Equal Set14.Equal Set15.Non – equivalent Set16.Equal Set17.Equal Set

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

Answers:18.True (if n=1, then 2n+1=3, integer: if n=2, 2n+1=5, integer: so the statement if

true)19.False ( if x=0 then cos0=1, sin0=0: so the statement is false)20.True (if even number is 2, & odd number is 1 than the sum will 3: which is

integer. So the statement is true)21.True

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

Answers: 7 (5.7+6.6+7.2+9.3+6.2=35: 35/5=7)

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.

Answers:23.Mean=12.5(sum of marks=150, mean 150/12)

24.Mean=15.5( sum of marks=186, mean 186/12)25.Mean=24.17( sum of marks=290, mean290/12)

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

Answer: a) always even

27. A vertex of degree one is called as a) Pendant b) isolated vertex c) null vertex d) colored vertex

Answer: a) pendant

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

Answer: b) Universal

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

Answer: b) even degree

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

Answer: a) n-1

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

Answer: d) 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

Answer: c) contains a circuit

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

Answers: b) 2

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

a) 2n-1 b) 2n

c) nn-1 d) nn

Answers: c) nn-1

35. The number of circuit in a tree with n nodes a) zero b) 1 b) n-1 d) n/2

Answer: 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

Answer 36 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

Answer: a) (z,*) is a monoid

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

38. Z, where * is subtraction

Answer: False

39. R, where a * b = a + b + 2 Answer: False

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

Answer: True