Define: i) Data and information ii) Abstract Data Types iii) Polynomials

I. Definition of Data and Information

A. Explanation of Data

Data can be defined as raw, unprocessed facts and statistics collected together for reference or analysis. They are the basic elements that provide a basis for reasoning, discussion, or calculation. Data can exist in various forms, such as numbers, text, images, etc. For instance, the temperature readings collected over a week, the words in a document, or the pixels in a photo, all constitute data.

B. Explanation of Information

Information, on the other hand, is data that has been processed to be meaningful. It is the result of processing, manipulating, and organizing data in a way that adds context, relevance, and purpose. For example, if the collected temperature readings are analyzed to find an average, or the words in a document are read and understood, or the pixels in a photo are viewed as a coherent image, they become information.

C. Difference between Data and Information

Both data and information are crucial in data structures. Data structures are used to organize and store data, while algorithms are used to process this data into meaningful information.

II. Definition of Abstract Data Types (ADTs)

A. Explanation of ADTs

Abstract Data Types (ADTs) can be defined as a high-level description of how data is viewed and the operations that can be performed on it, without regard to how they will be implemented. This concept is known as data encapsulation, where the details of implementation are hidden and only the necessary operations are exposed.

B. Examples of ADTs

Common examples of ADTs include:

• List: An ordered collection of items where each item holds a relative position.
• Stack: A collection of items where the addition of new items and the removal of existing items always takes place at the same end (top of the stack).
• Queue: A collection of items where the addition of new items happens at one end (rear) and the removal of existing items occurs at the other end (front).
• Tree: A collection of items (nodes) arranged in a hierarchical structure.
• Graph: A collection of items (vertices) where pairs of vertices are connected by edges.

ADTs provide a way to manage the complexity of data structures and algorithms by encapsulating the details of data and exposing only the necessary operations.

III. Definition of Polynomials

A. Explanation of Polynomials

Polynomials can be defined as an expression consisting of variables and coefficients, involving operations of addition, subtraction, multiplication, and non-negative integer exponents. A polynomial can be represented as: a_n*x^n + a_(n-1)*x^(n-1) + ... + a_2*x^2 + a_1*x + a_0, where a_n, a_(n-1), ..., a_2, a_1, a_0 are coefficients, and x is a variable.

B. Examples of Polynomials

Examples of polynomials include:

• 2x^2 + 3x + 1
• 5x^3 - 4x^2 + 7x - 8
• 6x - 7

The degree of a polynomial is the highest power of the variable in the polynomial. For instance, the degree of the polynomial 2x^2 + 3x + 1 is 2.

C. Polynomials in Data Structures

Polynomials can be represented and manipulated in data structures. For example, a polynomial can be represented as an array or a linked list where each node or element represents a term of the polynomial.

Here is a simple program in C++ to represent a polynomial using an array:

#include
using namespace std;

class Polynomial {
    int degree;
    int *coeff;

public:
    Polynomial(int degree) {
        this->degree = degree;
        coeff = new int[degree+1];
    }

    void setCoefficients() {
        cout << "Enter coefficients: ";
        for(int i=0; i<=degree; i++)
            cin >> coeff[i];
    }

    void display() {
        for(int i=0; i<=degree; i++) {
            cout << coeff[i] << "x^" << i;
            if(i != degree)
                cout << " + ";
        }
    }
};

int main() {
    int degree;
    cout << "Enter degree of polynomial: ";
    cin >> degree;

    Polynomial p(degree);
    p.setCoefficients();

    cout << "Polynomial: ";
    p.display();

    return 0;
}

This program creates a Polynomial class that represents a polynomial of a given degree. The coefficients of the polynomial are stored in an array. The setCoefficients() method is used to input the coefficients, and the display() method is used to display the polynomial.

Summary

Data and information are two related but distinct concepts. Data refers to raw, unprocessed facts and statistics, while information is data that has been processed and organized to be meaningful. Abstract Data Types (ADTs) are high-level descriptions of how data is viewed and the operations that can be performed on it, without regard to implementation details. Polynomials are expressions consisting of variables and coefficients, involving operations of addition, subtraction, multiplication, and non-negative integer exponents.

Analogy

Data is like a pile of unsorted puzzle pieces, while information is the completed puzzle. ADTs are like blueprints for building different types of houses, where the specific materials and construction methods are not specified. Polynomials are like mathematical equations that can represent various real-world phenomena, such as the trajectory of a projectile or the growth of a population.