Question

Create class Integerset for which each object can hold integers in the range 0 through \(100 .\) A set is represented internally as an array of ones and zeros. Array element al i 1 is 1 if integer \(i\) is in the set. Array element al \(j\) ] is 0 if integer \(j\) is not in the set. The default constructor initializes a set to the socalled "empty set," i.e., a set whose array representation contains all zeros. Provide member functions for the common set operations. For example, provide a unionofsets member function that creates a third set that is the settheoretic union of two existing sets (i.e., an element of the third set's array is set to 1 if that element is 1 in either or both of the existing sets, and an element of the third set's array is set to 0 if that element is 0 in each of the existing sets). Provide an intersectionofsets member function which creates a third set which is the set-theoretic intersection of two existing sets (i.e., an element of the third set's array is set to 0 if that element is 0 in either or both of the existing sets, and an element of the third set's array is set to 1 if that element is 1 in each of the existing sets). Provide an insertElement member function that inserts a new integer \(k\) into a set (by setting al \(k\) ) to 1 ). Provide a deleteElement member function that deletes integer \(m\) (by setting al \(m\) ) to 0 ). 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 array has a value of 1). Print \(\ldots\) 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 rntegerset class. Instantiate several Integerset objects. Test that all your member functions work properly.

Short answer

Define and implement an IntegerSet class with various methods for set operations and test using a driver program.

Step-by-step solution

Define the IntegerSet Class and Attributes

The `IntegerSet` class should contain an array of size 101 to represent possible integers from 0 to 100. This array, typically of integers, will be initialized to zeros by the default constructor to represent an empty set. Each index of the array corresponds directly to its value in the set. For instance, if index 4 has a value of 1, then the integer 4 is part of the set.

Implement the Default Constructor

Create a constructor for the `IntegerSet` class that initializes the array to zeros. This will ensure that any newly created set starts empty. Use a loop to iterate over each element of the array, setting each one to 0.

Implement the Union of Sets Method

This method should take another `IntegerSet` as input, and produce a new `IntegerSet` that is the union of the two sets. Loop through each index of the array, and set it to 1 if either of the corresponding indices in the two sets is 1.

Implement the Intersection of Sets Method

This method should also take another `IntegerSet` as input, and produce a new `IntegerSet` that is the intersection of the two sets. Loop through each index, setting it to 1 only if both corresponding indices in the two sets are 1.

Implement the Insert Element Method

To insert an element into the set, set the corresponding index to 1. For instance, to insert integer `k`, access the array at index `k` and assign it a value of 1. Before setting, verify that `k` is within the valid range of 0 to 100.

Implement the Delete Element Method

To delete an element from the set, set the corresponding index to 0. Access the array at index `m`, and assign it a value of 0, ensuring that `m` falls within the allowed range of 0 to 100 before proceeding.

Implement the Print Set Method

This method should go through the array and print numbers whose corresponding index in the array is 1, separated by spaces. If no indices are 1, print "..." to signify an empty set.

Implement the isEqualTo Method

Compare two `IntegerSet` objects by checking each index of their arrays. If all corresponding indices in both the arrays are equal, the sets are equal. Otherwise, they are not equal.

Implement the Additional Constructor

Create an additional constructor that receives an array of integers and its size. Initialize the `IntegerSet` by setting the array indices to 1 for each integer present in the input array, ensuring each integer is within the range 0-100.

Write a Driver Program

Write a separate program to create multiple `IntegerSet` objects. Use this program to test each method: `unionOfSets`, `intersectionOfSets`, `insertElement`, `deleteElement`, `printSet`, and `isEqualTo`, ensuring they function as expected.

Key concepts

Set Operations

Set operations allow us to manipulate sets and perform calculations such as union and intersection.
These operations are fundamental concepts in mathematics and computer science.
In the context of an `IntegerSet` class, they become concrete methods that offer logical handling of integer collections.

Here's how it works in the `IntegerSet` class:

• **Union of Sets**: This operation combines two sets into one, containing all unique elements from both. The `unionOfSets` method in the `IntegerSet` class iterates over each element index and flips the index to 1 if it was 1 in either of the operand sets. This effectively captures all elements present in at least one of the input sets.


• **Intersection of Sets**: This operation extracts only the common elements from two sets. The `intersectionOfSets` method in the `IntegerSet` class ensures that the index is set to 1 only if it was 1 in both operand sets. This operation highlights shared members between the sets.


• **Insertion and Deletion**: Besides the combination operations, sets also require the ability to change membership. The `insertElement` and `deleteElement` methods handle adding/removing an integer. They modify the corresponding index in the array representation of the set.

Understanding these operations is crucial for effectively using the `IntegerSet` class and performing set manipulations in a programmatic environment.

Array Representation of Sets

The array representation of sets is an efficient technique to store and manipulate sets in programming, particularly when dealing with a known, limited range of elements.
In the `IntegerSet` class used here, a fixed-size array of 101 elements represents all possible integers from 0 to 100.
Each element of the array is either a 0 or 1.

**How It Works:**

• **Binary Indicators**: Every index in the array acts as a potential member of the set. If the element at a specific index has a value of 1, the corresponding integer is included in the set. For example, if `setArray[5]` is 1, then the integer 5 is considered part of the set.


• **Initialization to Empty**: The default construction of the set results in all indices set to 0, effectively representing an empty set.


• **Efficiency**: This approach is highly efficient for performing operations like union and intersection since these involve simple index comparisons and assignments.


• **Simplicity**: This straightforward model avoids complex data structures, making it suitable for easy understanding and quick computations, at the cost of having a large array if the range is big but the set is sparse.

This representation is particularly useful when the set size is fixed or when rapid membership testing and modification are required.

Class Constructors

Constructors in classes are special functions used to initialize objects.
They ensure that new objects are created in a valid initial state.
In the `IntegerSet` class, constructors play a vital role in setting up the array that represents the set.

**Two Types of Constructors:**

• **Default Constructor**: This initializes the `IntegerSet` to an empty set. In practical terms, this means that all positions in the array are set to 0. By looping through each element during initialization, it guarantees an unambiguous starting state of the set, ready for manipulation.


• **Parameterized Constructor**: Beyond initializing an empty set, the `IntegerSet` class provides a way to directly construct a set with certain integers from an initial input. This additional constructor receives an array of integers and their count, setting the corresponding indices of those integers to 1. This allows for flexibility in object creation, letting users quickly instantiate a set starting with predefined elements.

Constructors are key in object-oriented programming, providing the scaffolding for stable and predictable object creation.

C++ Programming

C++ is a powerful programming language that allows for low-level memory manipulation but still supports object-oriented programming constructs, like classes.
This makes it a versatile choice when implementing fundamental data structures and algorithms.

**Key Features Used in `IntegerSet` Class Implementation:**

• **Class Implementation**: C++ allows encapsulating data and behaviors in classes, making `IntegerSet` an illustrative example. It keeps related functionalities bundled together, promoting organized and modular code design.


• **Array Handling**: The language has robust support to work with arrays. The `IntegerSet` class uses a simple integer array to store set membership, allowing for straight-forward iteration and indexing necessary to perform set operations.


• **Functionality via Methods**: C++ supports defining methods within a class that can act on class data. In `IntegerSet`, methods like `unionOfSets` and `insertElement` demonstrate the integration of operations tied directly to object instances, showcasing code reusability and conciseness.


• **Constructor Use**: The ability to define constructors in C++ helps in ensuring objects are ready for use immediately upon creation. This feature aids in keeping the `IntegerSet` objects correctly initialized without manual setup by the programmer after creation.

C++'s blend of direct memory manipulation capabilities along with a robust object-oriented paradigm makes it ideal for implementing efficient and structured solutions like the `IntegerSet` class.