Mathematics Assignment, Study Guides, Projects, Research of Mathematics

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Ramesh Devkota (HND / Third Semester) [TINKUNE, KATHMANDU]

DISCRETE MATH

Contents

PART ONE

With reference to the scenario, prepare a report which examines the set theory and functions applicable to software engineering and analyze mathematical structures of objects using graph theory.

Section 1

• Perform algebraic set operations in a formulated mathematical problem.
  • Determine the cardinality of a given bag (multi set).
• Determine the inverse of a function using appropriate mathematical techniques.
• Formulate corresponding proof principles to prove properties about defined sets.

Section 2

• Model contextualized problems using trees, both quantitatively and qualitatively.
  • Use Dijkstra's algorithm to find a shortest path spanning tree in a graph.
  • Assess whether an Euler Ian and Hamiltonian circuit exists in an undirected graph.
• Construct a proof of the Five Color Theorem.

Introduction I have been working as a mathematical analyst for the Central Bureau of Statistics operating in Kathmandu Nepal. I am going to perform algebraic set operation to formulated mathematical problem with explain the concept of bag as well as determine the cardinality for the set. I am also going to determine the inverse of a function by using an appropriate mathematical techniques, formulates corresponding proof principles to prove properties about define sets and going to model contextualized problems using trees, both quantitatively and qualitatively. Then I will use Dijkstra’s algorithm to find a shortest path spanning tree in a graph and after that I will assess whether an Euler Ian and Hamiltonian circuit exists in an undirected graph. And at last I will construct a proof of the Five Color Theorem.

Set A group or collection of objects or numbers, considered as an entity unto itself is called set (Rouse, 2019). Sets are usually symbolized by uppercase, italicized, boldface letters such as A, B, S, or Z and

each object or number in a set is called a member or element of the set. Examples include the set of all the apples on the tress, the set of all irrational numbers between 0 and 1 and the set of all computers in the world. It is customary to enclose the list in curly brackets when the elements of a set can be listed or enumerated. Thus, for example, we might speak of the set (call it R) of all natural numbers between, and including, 5 and 10 as:

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

A set can have any non-negative quantity of elements, ranging from none (the empty set or null set) to infinitely many. The number of elements in a set is called the cardinality, and can range from zero to denumerable infinite (for the sets of natural numbers, integers, or rational numbers) to non- denumerable infinite for the sets of irrational numbers, real numbers, imaginary numbers, or complex numbers).

Union

The smallest set that contains all the elements of both the sets is known as union. To find the union of two given sets A and B is a set which consists of all the elements of A and all the elements of B such that no element is repeated.

The symbol for denoting union of sets is ‘ U ’.

For example;

Let set A = {1, 2, 3, 4} and set B = {3, 4, 5, 6}

Taking every element of both the sets A and B, without repeating any element, we get a new set = {1, 2, 3, 4, 5, 6}

This new set contains all the elements of set A and all the elements of set B with no repetition of elements and is named as union of set A and B.

Intersection

The elements that are common in two sets are known as intersections.If an element is in just one set it is not part of the intersection. The intersection of two sets A and B, denoted by A ∩ B, is the set containing all elements of A that also belong to and nothing else.

Let set A = {1, 2, 3} and set B = {5, 7, 9}

A and B are disjoint set both of them have no common elements.

Complement Before we define complement of a set, we should define universal set and subset. A universal set is the set of all elements that are under consideration for a particular problem or situation. We have a set A that is a subset of some universal set U. The complement of A is the set of elements of the universal set that are not elements of A.

Given a set A, the complement of A is the set of all element in the universal set U, but not in A.

We can write Ac

we can also say complement of A in U

For example; Let set U = { a, b, c, d, f} And set A = {a, b, c} Therefore Ac^ = {d, f}

Properties of Union and intersection of sets  Associative Properties: A ∪ (B ∪ C) = (A ∪ B) ∪ C and A ∩ (B ∩ C) = (A ∩ B) ∩ C  Commutative Properties: A ∪ B = B ∪ A and A ∩ B = B ∩ A  Identity Property for Union: A ∪φ = A  Intersection Property of the Empty Set: A ∩φ = φ  Distributive Properties: A ∪(B ∩ C) = (A ∪ B) ∩ (A ∪ C) similarly for intersection

Set complement: Absolute and Relative   The (absolute) complement of A is A=U\A  The (relative) complement of A in B is B\A

Bag

It is the essential like set object but it differ from as set in that it allows for multiple copies of the same element. Bags are conceptually unordered collection of elements with duplicates.

For Example;

U=< u, v, u, w, x, y, z>

U bag or multiset is a set in which duplicates are allowed but only a finite number of duplicates. Insteadof curly brackets we use square brackets to list the element of a bag.

U=[u, v, v, w]

(|U|=4)

V = [u, v, w, x, x]

(|V|=5)

Finding the cardinal number of a set The number of distinct elements in a finite set is called its cardinal number. It is denoted as n(A) and read as ‘the number of elements of the set’.

Examples:

a) A= {y: y is a planet in our solar system} Solution There are eight planets; therefore, this is a finite set. n (p) = b) B= {7,8, 9, A} Solution The set of counting number is an infinite set. c) C= {a : a is a person living in the Nepal who is not Nepali} Solution

By using Venn diagram a set can be represented graphically.

Function A function is a relation from a set of inputs to a set of possible outputs where each input is related to exactly one output. In the other word it is describe as a relation between two variables called the independent variable and the dependent variable. For any specific value of the independent variable, the dependent variable has at most one value. By a lowercase, italicized letter of the alphabet a function is usually symbolized and followed by the independent variable in parentheses.

Example: with f(x) = x^2

Whereas (^) f = Function name

(x) =input X2 = what to output

Domain of Function

The domain of function is defined as the maximum set of values, typically within the reals but sometimes among the integers or complex numbers.

Given a function f:A B , the set A is the domain of f ; the set B is the codomain of f. In the expression f(a) , a is the argument and f(a) is the value. The values as the “output” when the function is applied to that member of the domain. An argument as member of the domain that is chosen as an “input” to the function.

Range of a function

The range of a function is the set of outputs the function achieves when it is applied to its whole set of outputs.

Range of a function b = f (a) is the same as finding all values that y could be. The range of f(x) is all the y-values where there is a number x with b=f (a).

For example: - f (a) = a^2 + 1

To find b -values such that there is some a where b = a^2 + 1. Suppose we want to check if b=5 is in the range of f (a). Then, we want to check if there is an a-value such that a^2 + 1=5. We can solve this equation as follows:

a^2 + 1= a^2 = a= ±

So since either x=2 or x= -2 works, we know that y=5 is in the range of f(a).

Graphica l form

“A graph is commonly used to give an intuitive picture of a function. It is easy to see from its graph whether a function is increasing or decreasing. The function f: R - {0} → R is represented in the graph such that the x co-ordinate represents the independent variable and the y co-ordinate represents the dependent variable.” The limiting case of the graph of the function is represented by an asymptote:

The function approaches at infinity but does not exactly touch it as shown in the figure shown above.

Tabular form

0 from the set of first components there is only one number in the list of second components associated with each. This doesn’t matter. The fact that we found even a single value in the set of first components with more than one second component associated with it is enough to say that this relation is not a function. As a final comment about this example let’s note that if we removed the first and or the fourth ordered pair from the relation we would have a function

Terms related to functions:

 Domain and co-domain - if f is a function from set X to set Y, then A is called Domain and B is called co-domain.  Range - Range of f is the set of all images of elements of X. Basically Range is subset of co- domain.  Image and Pre-Image – Y is the image of X and X is the pre-image of Y if f(x)=y

Types of function:

One to one function (injective): An injective function is a function that maps distinct elements of its domain to distinct elements of its codomain.

Onto function:

If every element y in Y has a corresponding element x in X such that f(x)=y. The function f may map one or more elements of X to the same element of Y. It is not required that x is unique.

Set A and Set B which consist of elements. Onto function could be explained by considering two sets. If for every element of B there is at least one or more than one element matching with A then the function is said to be onto function or surjective function.

One to one correspondence function (invertible/bijective):

If a function f: A B satisfies both the injective and surjective function properties is known as one to one correspondence function. A function f: A → B is a bijective function if every element b ∈ B

and every element a ∈ A, such that f(a) = b.

Invertible Function:

A function f from a set X to a set Y is said to be invertible if there exists a function g from Y to X such that f(g(y)) = y and g(f(x)) = x for every y in Y and x in X. Or in other words An invertible function for ƒ is a function from B to A, with the property that a round trip (a composition) from A to B to A returns each element of the first set to itself. A function ƒ that has an inverse is called invertible; the inverse function is then uniquely determined by ƒ and is denoted by ƒ^-1. For example 1: Let A = {1, 2, 3}, B = {a, b, c, d}. Consider a function f = {(1, a), (2, b), (3, c)}. Here the image set or the range is {a, b, c} which is not equal to the co domain {a, b, c, d}. Therefore, it is not onto. For the inverse function f−1 the co-domain off becomes domain of f −1. I.e. If f: A → B then f−1 : B

the two.

Find the inverse of y = –^2 / (^) ( x – 5), and determine whether the inverse is also a function.

Since the variable is in the denominator, this is a rational function. Here's the algebra:

The original function: I multiply the denominator up to the left-hand side of the equation: I take the y through the parentheses: I get the x -stuff by itself on one side of the "equals" sign: Then I solve for x : And then switch the x 's and y 's:

This is just another rational function. The inverse function is y = (^5 x^ –^ 2)^ / (^) x How to find the inverse of a function

1. First, replace f(x) with y.

2. Replace every x with a y and replace every y with an x.
3. Solve the equation from Step 2 for y.
4. Replace y with f−1(x) f − 1 (x).
5. Verify your work by checking that (f∘f−1)(x)=x ( f ∘ f − 1 ) ( x ) = x and (f−1∘f)(x)=x ( f − 1 ∘ f ) ( x ) = x are both true.

Boolean identities

Commutativity The commutative property in math comes from the words "commute" or "move around." This rule states that you can move numbers or variables in algebra around and still get the same answer.

Proving through logic gate

Proving through truth table

Associativity A calculation that gives the same result regardless of the way the numbers are grouped is called associativity.

Proving through logic gate

Truth table