Question

\( \quad\) (Integerset Class) Create class IntegerSet for which each object can hold integers in the range 0 through \(100 .\) Represent the set internally as a vector of bool values. Element a \([\mathrm{i}]\) is true if integer \(i\) is in the set. Element a \([j]\) is \(f\) alse if integer \(j\) is not in the set. The default constructor initializes a set to the so-called "empty set," i.e., a set for which all elements contain false. Provide member functions for the common set operations. For example, provide a unionofSets member function that creates a third set that is the set-theoretic union of two existing sets (i.e., an element of the result is set to true if that element is true in either or both of the existing sets, and an element of the result is set to false if that element is false in each of the existing sets). Provide an intersectionofsets member function which creates a third set which is the settheoretic intersection of two existing sets (i.e., an element of the result is set to false if that element is \(f\) al se in either or both of the existing sets, and an element of the result is set to true if that element is true in each of the existing sets). Provide an insertE7ement member function that places a new integer \(k\) into a set by setting \(\mathrm{a}[\mathrm{k}]\) to true. Provide a de 7 ete \(\mathrm{E}\) 1 ement member function that deletes integer \(m\) by setting a \([\mathrm{m}]\) to false. Provide a printset member function that prints a set as a list of numbers separated by spaces. Print only those elements that are present in the set (i.e., their position in the vector has a value of true). Print \(--\) for an empty set. Provide an isEqualTo member function that determines whether two sets are equal. Provide an additional constructor that receives an array of integers and the size of that array and uses the array to initialize a set object. Now write a driver program to test your Integerset class. Instantiate several Integerset objects. Test that all your member functions work properly.

Short answer

Define IntegerSet with a vector of bools; implement member functions for union, intersection, insertion, deletion, printing, and equality checks; provide additional constructor; test with a driver program.

Step-by-step solution

Define the IntegerSet Class

Define a class named IntegerSet. Include a private member vector of boolean values of size 101 to represent the presence of integers between 0 and 100 inclusively. Add a default constructor to initialize all elements in the vector to false, representing an empty set.

Implement Member Functions

Implement the following member functions:1. unionOfSets: Takes another IntegerSet object as an argument and returns a new IntegerSet which is the union of the current set and the passed set.2. intersectionOfSets: Takes another IntegerSet object and returns a new IntegerSet which is the intersection of the current set and the passed set.3. insertElement: Takes an integer k and sets the k-th element of the vector to true.4. deleteElement: Takes an integer m and sets the m-th element of the vector to false.5. printSet: Prints numbers separated by spaces for which the corresponding element in the vector is true and prints '--' for an empty set.6. isEqualTo: Compares the current set with another IntegerSet object to determine if they are equal.Provide proper implementation details within each member function.

Provide Additional Constructor

Provide an additional constructor for the IntegerSet class which takes an array of integers and the size of the array. The constructor should use the array to set the corresponding indices in the internal vector to true.

Write a Driver Program

Write a driver program (main function) to instantiate several IntegerSet objects and test all the member functions. Ensure you include checks for union, intersection, insertion, deletion, printing, equality comparison, and initialization from an array. Confirm by visual inspection or programmatically that each operation provides the expected output.

Key concepts

C++ Class Implementation

In the C++ programming language, a class is a blueprint for creating objects which encapsulate data and operations on data.
A common practice in C++ class implementation is to use private members to protect data, often through the use of accessor and mutator functions. In our IntegerSet class exercise, a private member is a 'vector' sized to contain values representing the presence or absence of integers within a given range. The default constructor set all indices of this 'vector' to 'false', indicating an empty set.

Member functions, or methods, within the class are designed to handle specific tasks relevant to the 'set theory', such as adding or removing elements (insertElement and deleteElement), comparing sets (isEqualTo), and printing the set out in a human-readable form (printSet). Each operation changes the state of the set in some manner, demonstrating encapsulation and data management fundamental to a well-designed class in C++.

The C++ standard library provides support for exception handling, constructors, destructors, and the RAII (Resource Acquisition Is Initialization) principle to ensure resources are properly managed. Implementing a class like IntegerSet in C++ requires not only the handling of boolean elements but also careful consideration of these programming concepts for robust and efficient code.

Set Theory Operations

Set theory is a mathematical theory that deals with the properties and relations of sets, which are collections of objects. In the IntegerSet class, we express set theory through Boolean vectors and implement operations such as union and intersection, fundamental to understanding set theory.

The unionOfSets function represents the set union operation, which combines two sets to form a new set containing all elements that are present in either set. Mathematically, if set A and set B are two sets, the union is denoted as A ∪ B. In the context of our class, the union is a new IntegerSet with elements set to 'true' if they appear in either of the original sets.

The intersectionOfSets function implements set intersection, which results in a new set containing only elements that are present in both sets. Symbolically, the intersection of set A and set B is written as A ∩ B. For the IntegerSet class, this translates to an IntegerSet with elements set to 'true' only if both original sets had the corresponding elements set to 'true'.

By applying these operations, students learn the practical application of set theory in programming, reinforcing concepts such as the complement and difference of sets, subsets, and the cardinality of a set in a more concrete and tangible way using code.

C++ Vectors

Vectors in C++ are part of the Standard Template Library (STL) and provide a way to store and manage a collection of objects. They are similar to arrays but have the advantage of being able to dynamically resize themselves to accommodate additional elements.

In the IntegerSet example, we utilize a vector of booleans to represent the presence (true) or absence (false) of each possible integer in the set from 0 to 100. The choice of a vector instead of a regular array offers the benefits of automatic memory management and easy access to high-level operations such as iterators, which are useful when implementing functions like printSet to iterate through the set and display its contents.

The dynamically resizable nature of vectors means functions like insertElement can add elements without concern for the capacity of the underlying array. When used correctly, this can result in safer and more efficient code, as vectors can manage underlying storage and optimize allocations to minimize overhead. Moreover, vectors come with utility functions for checking size, testing if they're empty, and accessing elements, which provide a programmer-friendly interface for managing collections of data in C++.