To support sets, the .NET Framework provides a class named HashSet. This class is created in the System.Collections.Generic namespace.

In algebra, a set is usually represented with an uppercase letter, such as A. In programming, we will represent the list with a name.

Consider the following list and notice that it contains an item twice:

using System;
using System.Windows.Forms;
using System.Collections.Generic;

// Use references to
// System
// System.Drawing
// System.Windows.Forms

public class SetHasher : Form
{
    ListBox lbxNames;

    public SetHasher()
    {
        InitializeComponent();
    }

    private void InitializeComponent()
    {
        lbxNames = new ListBox();
        lbxNames.Location = new System.Drawing.Point(12, 12);
        lbxNames.Size = new System.Drawing.Size(120, 95);
        StartPosition = FormStartPosition.CenterScreen;

        Text = "Set Hasher";
        MinimizeBox = false;
        MaximizeBox = false;
        ClientSize = new System.Drawing.Size(145, 120);
        Load += new EventHandler(FormLoaded);
        Controls.Add(lbxNames);
    }

    private void FormLoaded(object sender, EventArgs ea)
    {
        List<string> names = new List<string>();

        names.Add("Kevin");
        names.Add("Jane");
        names.Add("Daniel");
        names.Add("Ruth");
        names.Add("Jane");
        names.Add("Paul");

        lbxNames.DataSource = names;
    }
}

public class Exercise
{
    [STAThread]
    public static int Main()
    {
        Application.Run(new SetHasher());
        return 0;
    }
}

In a .NET application, a set is a list that does not allow duplicate items.

Most of the time, in algebra, to study sets, we use numbers. In all of the examples in this lesson, we will use strings to make them more interesting. Obviously, integers would be easier. If you want to use numbers, all you have to do is to replace all strings in our examples with numbers.

1. Start Microsoft Visual Studio or Microsoft Visual C# Express
2. Create a Windows Application named Algebra2
3. In the Solution Explorer, right-click Form1.cs and click Rename
4. Type Algebra.cs and press Enter
5. Click the body of the form to select it
6. In the Properties window, change the following characteristics
   FormBorderStyle: FixedDialog
   Text: Operations on Sets
   StartPosition: CenterScreen
   MaximizeBox: False
   MinimizeBox: False
7. Design the form as follows: 
    

   Control Text Name Label Number:   TextBox   txtNumberA Button Add btnAddA ListBox   lbxA

In algebra, a set is empty if it contains no element. This is the case for a brand new set. It can also happen if the elements in the set have been deleted. In algebra, to create an empty set, use an uppercase letter and assign the phi character to it. Here is an example:

A = ∅

(In algebra, an empty set is also called a null set.) In algebra, to visually represent a set, we draw a circle. If the set is empty, the circle is blank:

To programmatically create an empty set, declare a variable of type HashSet and use the new operator to initialize it.

Depending on the type of set, one set can be made of natural numbers, another made of floating-point numbers, yet another one made of strings or other types of values (dates, times, etc). When programmatically creating a set, pass the type of items in the angle brackets. Here is an example:

private void FormLoaded(object sender, EventArgs ea)
{
    HashSet<string> names = new HashSet<string>();
}

The HashSet class is defined as follows:

[SerializableAttribute]
[HostProtectionAttribute(SecurityAction.LinkDemand, MayLeakOnAbort = true)]
public class HashSet<T> : ICollection<T>, IEnumerable<T>, 
    IEnumerable, ISerializable, IDeserializationCallback

1. Double-click an unoccupied of the form to generate its Load event
2. Change the file as follows:

   using System;
   using System.Collections.Generic;
   using System.ComponentModel;
   using System.Data;
   using System.Drawing;
   using System.Linq;
   using System.Text;
   using System.Windows.Forms;
   
   namespace Algebra2
   {
       public partial class Algebra : Form
       {
           HashSet<int> A;
   
           public Algebra()
           {
               InitializeComponent();
           }
   
           private void Algebra_Load(object sender, EventArgs e)
           {
               A = new HashSet<int>();
           }
       }
   }

3. Return to the form

Unless a set is empty, it must have or contain something. An object included in a set is called an element or a member of the set. In algebra, to create a set, we specify an uppercase letter, followed by = { }. In the curly brackets, we list the elements separated by commas. Here is an example:

A = { Kevin, Jane, Daniel, Ruth, Paul }

To visually illustrate, you can represent each element in the circle. Here is an example:

In programming, to create a set, after declaring a HashSet variable, we can call its Add() method whose syntax is:

public bool Add(T item);

Here are examples of calling this method:

private void FormLoaded(object sender, EventArgs ea)
{
    HashSet<string> names = new HashSet<string>();

    names.Add("Kevin");
    names.Add("Jane");
    names.Add("Daniel");
    names.Add("Ruth");
    names.Add("Paul");

    foreach (string str in names)
        lbxNames.Items.Add(str);
}

Notice that each element in this case is a constant, or has a constant value.

Remember that one of the rules of a set is that it cannot have duplicate items. If you add an item that exists in the list already, that item would not be added. In fact, to let you know whether an item was added, the HashSet<>.Add() method returns a Boolean value:

• If the item is successfully added, the method returns true
• If the addition is denied, the method returns false and the compiler does not throw an exception

The following code will illustrate:

private void FormLoaded(object sender, EventArgs ea)
{
    HashSet<string> names = new HashSet<string>();

    if (names.Add("Kevin") == true)
        MessageBox.Show("Kevin has been added to the list.",
                        "Hash Set Operations",
                        MessageBoxButtons.OK, MessageBoxIcon.Information);
    else
        MessageBox.Show("Kevin was not added to the list.",
                        "Hash Set Operations",
                        MessageBoxButtons.OK, MessageBoxIcon.Information);
    if (names.Add("Jane") == true)
        MessageBox.Show("Jane has been added to the list.",
                        "Hash Set Operations",
                        MessageBoxButtons.OK, MessageBoxIcon.Information);
    else
        MessageBox.Show("Jane was not added to the list.",
                        "Hash Set Operations",
                        MessageBoxButtons.OK, MessageBoxIcon.Information);
    if (names.Add("Daniel") == true)
        MessageBox.Show("Daniel has been added to the list.",
                        "Hash Set Operations",
                        MessageBoxButtons.OK, MessageBoxIcon.Information);
    else
        MessageBox.Show("Daniel was not added to the list");
    if (names.Add("Ruth") == true)
        MessageBox.Show("Ruth has been added to the list.",
                        "Hash Set Operations",
                        MessageBoxButtons.OK, MessageBoxIcon.Information);
    else
        MessageBox.Show("Ruth was not added to the list.",
                        "Hash Set Operations",
                        MessageBoxButtons.OK, MessageBoxIcon.Information);
    if (names.Add("Jane") == true)
        MessageBox.Show("Jane has been added to the list.",
                        "Hash Set Operations",
                        MessageBoxButtons.OK, MessageBoxIcon.Information);
    else
        MessageBox.Show("Jane was not added to the list.",
                        "Hash Set Operations",
                        MessageBoxButtons.OK, MessageBoxIcon.Information);
    if (names.Add("Paul") == true)
        MessageBox.Show("Paul has been added to the list.",
                        "Hash Set Operations",
                        MessageBoxButtons.OK, MessageBoxIcon.Information);
    else
        MessageBox.Show("Paul was not added to the list.",
                        "Hash Set Operations",
                        MessageBoxButtons.OK, MessageBoxIcon.Information);

    foreach (string str in names)
        lbxNames.Items.Add(str);
}

1. On the form, double-click the Add button
2. Implement its event as follows:

   private void btnAddNumbers_Click(object sender, EventArgs e)
   {
       int x = 0;
   
       try
       {
           x = int.Parse(txtNumber.Text);
           A.Add(x);
       }
       catch (FormatException)
       {
           MessageBox.Show("The value you typed is not a valid number.",
                           "Algebra of Sets",
                           MessageBoxButtons.OK, MessageBoxIcon.Information);
       }
   
       lbxNumbers.Items.Clear();
   
       foreach (int element in A)
           lbxNumbers.Items.Add(element);
   
       txtNumber.Text = "";
       txtNumber.Focus();
   }

3. Press Ctrl + F5 to execute
4. Type a number in the text box and click Add
5. Do that a few times
    
   
6. Close the form and return to your programming environment

In algebra, when it comes to the number of elements in a set, there are two categories. A set is referred to as finite it contains a constant number of items. An example is the number of students in a classroom. A set is infinite if its number of items cannot be counted. An example is the number of stars in the sky.

The HashSet class implements the ICollection<> generic interface, which gives it a property named Count. This property allows you to know the number of elements in a set. Here is an example:

using System;
using System.Windows.Forms;
using System.Collections.Generic;

public class SetHasher : Form
{
    Label lblCount;
    ListBox lbxNames;

    public SetHasher()
    {
        InitializeComponent();
    }

    private void InitializeComponent()
    {
        lbxNames = new ListBox();
        lbxNames.Location = new System.Drawing.Point(12, 12);
        lbxNames.Size = new System.Drawing.Size(120, 95);

        lblCount = new Label();
        lblCount.AutoSize = true;
        lblCount.Location = new System.Drawing.Point(12, 115);

        Text = "Set Hasher";
        MinimizeBox = false;
        MaximizeBox = false;
        StartPosition = FormStartPosition.CenterScreen;
        ClientSize = new System.Drawing.Size(145, 140);
        Load += new EventHandler(FormLoaded);

        Controls.Add(lbxNames);
        Controls.Add(lblCount);
    }

    private void FormLoaded(object sender, EventArgs ea)
    {
        HashSet<string> names = new HashSet<string>();

        names.Add("Kevin Walchild");
        names.Add("Jane Overton");
        names.Add("Daniel Albertson");
        names.Add("Ruth Oyawale");
        names.Add("Jane Ouelette");
        names.Add("Paul Sullivan");

        foreach (string str in names)
            lbxNames.Items.Add(str);

        lblCount.Text = "The list contains " + names.Count.ToString() + " items";
    }
}

public class Exercise
{
    [STAThread]
    public static int Main()
    {
        Application.Run(new SetHasher());
        return 0;
    }
}

In algebra, to represent that a certain value "a" exists in a set A, we would write:

a ∈ A

This is read as "a is an element of set A" or "a is included in set A" or "Set A contains a". To support this operation, the HashSet class inherits the Contains() method from the ICollection<> interface. The method allows you to check whether the list already contains a certain item. Here are examples of calling this method:

private void FormLoaded(object sender, EventArgs ea)
{
    HashSet<string> names = new HashSet<string>();

    names.Add("Kevin Walchild");
    names.Add("Jane Overton");
    names.Add("Daniel Albertson");
    names.Add("Ruth Oyawale");
    names.Add("Jane Ouelette");
    names.Add("Paul Sullivan");

    foreach (string str in names)
        lbxNames.Items.Add(str);

    if( names.Contains("Jane Owerton") )
        MessageBox.Show("The list contains Jane Owerton.",
                        "Hash Set Operations",
                        MessageBoxButtons.OK, MessageBoxIcon.Information);
    else
        MessageBox.Show("The list doesn't contain Jane Owerton",
                        "Hash Set Operations",
                        MessageBoxButtons.OK, MessageBoxIcon.Information);

    if( names.Contains("Jane Overton") )
        MessageBox.Show("The list contains Jane Overton.",
                        "Hash Set Operations",
                        MessageBoxButtons.OK, MessageBoxIcon.Information);
    else
        MessageBox.Show("The list doesn't contain Jane Overton",
                        "Hash Set Operations",
                        MessageBoxButtons.OK, MessageBoxIcon.Information);

    lblCount.Text = "The list contains " + names.Count.ToString() + " items";
}

In algebra, to state that an element a is not in the set A, we write:

a ∉ A

This reads "a is not an element of set A" or "a is not included in set A" or "Set A does not contain a". To perform this checking in programming, you can use the NOT operator "!".

In algebra, after creating a set A, you may want to create another set B but that contains the same elements as A. To do this, you can simply assign the elements of A to B. This would be done as follows:

A = { Kevin Walchild, Jane Overton, Daniel Albertson,
      Ruth Oyawale, Jane Ouelette, Paul Sullivan }

B = A

Now, sets A and B have the same elements:

To support this operation, the HashSet class provides another constructor (in C++ and Pascal, we could use a copy constructor) whose syntax is:

public HashSet(IEnumerable<T> collection);

This constructor allows you to create a new set that is made of values from an existing set. You create the new list by passing the name of the existing list to this constructor. Here is an example:

 private void FormLoaded(object sender, EventArgs ea)
{
        HashSet<string> names = new HashSet<string>();

        names.Add("Kevin Walchild");
        names.Add("Jane Overton");
        names.Add("Daniel Albertson");
        names.Add("Ruth Oyawale");
        names.Add("Jane Ouelette");
        names.Add("Paul Sullivan");

        HashSet<string> people = new HashSet<string>(names);
}

Once you have created the list, you can manipulate it as you see fit. For example, you can add new items to it. Here is an example:

using System;
using System.Windows.Forms;
using System.Collections.Generic;

public class SetHasher : Form
{
    Label lblCount;
    Label lblPeople;
    ListBox lbxNames;
    ListBox lbxPeople;

    public SetHasher()
    {
        InitializeComponent();
    }

    private void InitializeComponent()
    {
        lbxNames = new ListBox();
        lbxNames.Location = new System.Drawing.Point(12, 12);
        lbxNames.Size = new System.Drawing.Size(120, 120);

        lbxPeople = new ListBox();
        lbxPeople.Location = new System.Drawing.Point(138, 12);
        lbxPeople.Size = new System.Drawing.Size(120, 120);
        
        lblCount = new Label();
        lblCount.AutoSize = true;
        lblCount.Location = new System.Drawing.Point(12, 126);

        lblPeople = new Label();
        lblPeople.AutoSize = true;
        lblPeople.Location = new System.Drawing.Point(138, 126);

        Text = "Set Hasher";
        MinimizeBox = false;
        MaximizeBox = false;
        StartPosition = FormStartPosition.CenterScreen;
        ClientSize = new System.Drawing.Size(270, 150);
        Load += new EventHandler(FormLoaded);

        Controls.Add(lbxNames);
        Controls.Add(lbxPeople);
        Controls.Add(lblCount);
        Controls.Add(lblPeople);
    }

    private void FormLoaded(object sender, EventArgs ea)
    {
        HashSet<string> names = new HashSet<string>();

        names.Add("Kevin Walchild");
        names.Add("Jane Overton");
        names.Add("Daniel Albertson");
        names.Add("Ruth Oyawale");
        names.Add("Jane Ouelette");
        names.Add("Paul Sullivan");

        foreach (string str in names)
            lbxNames.Items.Add(str);

        lblCount.Text = "Number of items: " + names.Count.ToString();

        HashSet<string> people = new HashSet<string>(names);
        people.Add("Antoinette Belton");
        people.Add("Joshua Melmann");

        foreach (string str in people)
            lbxPeople.Items.Add(str);

        lblPeople.Text = "Number of items: " + people.Count.ToString();
    }
}

public class Exercise
{
    [STAThread]
    public static int Main()
    {
        Application.EnableVisualStyles();
        Application.SetCompatibleTextRenderingDefault(false);
        Application.Run(new SetHasher());
        return 0;
    }
}

1. Change the design of the form as follows: 
    

   Control Text Name Label Number:   TextBox   txtNumberA Button Add btnAddNumberA ListBox   lbxNumbersA Button Duplicate btnDuplicate ListBox   lbxNumbersB

2. Double-click the Duplicate button
3. Change the file as follows:

   using System;
   using System.Collections.Generic;
   using System.ComponentModel;
   using System.Data;
   using System.Drawing;
   using System.Linq;
   using System.Text;
   using System.Windows.Forms;
   
   namespace Algebra2
   {
       public partial class Algebra : Form
       {
           HashSet<int> A;
           HashSet<int> B;
   
           public Algebra()
           {
               InitializeComponent();
           }
   
           private void Algebra_Load(object sender, EventArgs e)
           {
               A = new HashSet<int>();
               B = new HashSet<int>();
           }
   
           private void btnAddNumberA_Click(object sender, EventArgs e)
           {
               int x = 0;
   
               try
               {
                   x = int.Parse(txtNumberA.Text);
                   A.Add(x);
               }
               catch (FormatException)
               {
                   MessageBox.Show("The value you typed is not a valid number.",
                                   "Algebra of Sets",
                                MessageBoxButtons.OK, MessageBoxIcon.Information);
               }
   
               lbxNumbersA.Items.Clear();
   
               foreach (int element in A)
                   lbxNumbersA.Items.Add(element);
   
               txtNumberA.Text = "";
               txtNumberA.Focus();
           }
   
           private void btnDuplicate_Click(object sender, EventArgs e)
           {
               B = A;
   
               foreach (int element in B)
                   lbxNumbersB.Items.Add(element);
           }
       }
   }

4. To execute, press Ctrl + F5
5. Add a few numbers to the left list box. Here is an example:
    
   
6. Click Duplicate
    
   
7. Close the form and return to your programming environment

Imagine you have two sets A and B defined as follows:

A = { Kevin Walchild, Jane Overton, Daniel Albertson, Jane Ouelette }
B = { Daniel Albertson, Kevin Walchild, Jane Ouelette, Jane Overton }

Notice that both sets have the same elements. When two sets have exact same members, we say that those sets are equal. This can be expressed as:

A = B

This operation is commutative, which means the above operation can also be written as:

B = A

We already know that all classes used in a C# application (or a .NET project) derive from the Object class and they inherit a method named Equals. This makes it possible to compare two values or two objects for equality. The HashSet class also naturally inherits this method. Although you can use that method, the HashSet class provides its own custom method named SetEquals. Its syntax is:

public bool SetEquals(IEnumerable<T> other);

This method takes as argument another HashSet list and compares both sets. If the sets are the same, the method returns true. Otherwise it produces a false value. Here are examples of calling this method:

private void FormLoaded(object sender, EventArgs ea)
{
    HashSet<string> names = new HashSet<string>();

    names.Add("Kevin Walchild");
    names.Add("Jane Overton");
    names.Add("Daniel Albertson");
    names.Add("Ruth Oyawale");
    names.Add("Jane Ouelette");
    names.Add("Paul Sullivan");

    foreach (string str in names)
        lbxNames.Items.Add(str);

    lblCount.Text = "Number of items: " + names.Count.ToString();

    HashSet<string> people = new HashSet<string>(names);

    if (people.SetEquals(names) == true)
        MessageBox.Show("Both collections are the same.",
                        "Collection Class",
                        MessageBoxButtons.OK, MessageBoxIcon.Information);
    else
        MessageBox.Show("These collections are different.",
                        "Collection Class",
                        MessageBoxButtons.OK, MessageBoxIcon.Information);

    people.Add("Antoinette Belton");
    people.Add("Joshua Melmann");

    if (people.SetEquals(names) == true)
        MessageBox.Show("Both collections are the same.",
                        "Collection Class",
                        MessageBoxButtons.OK, MessageBoxIcon.Information);
    else
        MessageBox.Show("These collections are different.",
                        "Collection Class",
                        MessageBoxButtons.OK, MessageBoxIcon.Information);

    foreach (string str in people)
        lbxPeople.Items.Add(str);

    lblPeople.Text = "Number of items: " + people.Count.ToString();
}

1. Add a button between the list boxes as follows: 
    

   Control Text Name Button Compare btnCompare

2. Double-click the Compare button
3. Implement the event as follows:

   private void btnCompare_Click(object sender, EventArgs e)
   {
       if (A.SetEquals(B))
           MessageBox.Show("A is equal to B",
                           "Algebra of Sets",
                           MessageBoxButtons.OK, MessageBoxIcon.Information);
       else
           MessageBox.Show("A is different from B",
                       "Algebra of Sets",
                       MessageBoxButtons.OK, MessageBoxIcon.Information);
   }

4. Return to the form
5. To execute, press Ctrl + F5
6. Add a few numbers to the left list box. Here is an example:
    
   
7. Click Compare
    
   
8. Click OK
9. Click Duplicate
10. Click Compare
     
    
11. Click OK
12. Close the form and return to your programming environment

Imagine you have two sets A and B defined as follows:

A = { Kevin, Jane, Daniel }
B = { Raul, Kevin, Tiffany, Daniel, Michael, Jane, Paul }

Notice that all members of A are also members of B. In algebra, to express this, we write:

A ⊂ B

This is the same as:

 { Kevin, Jane, Daniel } ⊂ { Raul, Kevin, Tiffany, Daniel, Michael, Jane, Paul }

We say that A is a subset of B. This can be visually illustrated as follows:

In algebra, any set is a subset of itself. Also, since the empty set doesn't have any element, the empty set is a subset of any set. We write it as:

∅ ⊂ A

and

∅ ⊂ B

An empty set is also a subset of itself.

To let you find out you whether one set is a subset of another, the HashSet class provides the IsSubsetOf() method. Its syntax is:

public bool IsSubsetOf(IEnumerable<T> other);

If the set passed as argument is a subset of the variable that called the method, it returns true. Otherwise, it returns false. Here is an example of calling this method:

private void FormLoaded(object sender, EventArgs ea)
{
    HashSet<string> Persons = new HashSet<string>();

    Persons.Add("Kevin Walchild");
    Persons.Add("Jane Overton");
    Persons.Add("Daniel Albertson");

    HashSet<string> People = new HashSet<string>();
    People.Add("Raul Fitzgerald");
    People.Add("Kevin Walchild");
    People.Add("Tiffany Morales");
    People.Add("Daniel Albertson");
    People.Add("Michael Rhodia");
    People.Add("Ruth Oyawale");
    People.Add("Jane Overton");
    People.Add("Paul Sullivan");

    if (Persons.IsSubsetOf(People))
        MessageBox.Show("The Persons set is a subset of the People set.",
                        "Collection Class",
                        MessageBoxButtons.OK, MessageBoxIcon.Information);
    else
        MessageBox.Show("The Persons set is not a subset of the People set.",
                        "Collection Class",
                        MessageBoxButtons.OK, MessageBoxIcon.Information);

    foreach (string str in Persons)
        lbxNames.Items.Add(str);

    lblCount.Text = "Number of items: " + Persons.Count.ToString();

    foreach (string str in People)
        lbxPeople.Items.Add(str);

    lblPeople.Text = "Number of items: " + People.Count.ToString();
}

It is important to know that the subset relationship is one way; in other words, the comparison is not commutative: the fact that a set A is a subset of a set B is not vice-versa.

1. Add another label to the form and change its Text to Number
2. Add a second text box and another button to the form
3. Change the design of the form as follows: 
    

   Control Text Name Label Number:   TextBox   txtNumberA Button Add btnAddNumberA Label Number:   TextBox   txtNumberB Button Add btnAddNumberB

4. On the form, double-click the second Add button
5. Implement its event as follows:

   private void btnAddNumberB_Click(object sender, EventArgs e)
   {
       int x = 0;
   
       try
       {
           x = int.Parse(txtNumberB.Text);
           B.Add(x);
       }
       catch (FormatException)
       {
           MessageBox.Show("The value you typed is not a valid number.",
                           "Algebra of Sets",
                           MessageBoxButtons.OK, MessageBoxIcon.Information);
       }
   
       lbxNumbersB.Items.Clear();
   
       foreach (int element in B)
           lbxNumbersB.Items.Add(element);
   
       txtNumberB.Text = "";
       txtNumberB.Focus();
   }

6. Change the code of the Compare button as follows:

   private void btnCompare_Click(object sender, EventArgs e)
   {
       if (A.SetEquals(B))
           MessageBox.Show("A is equal to B.",
                           "Algebra of Sets",
                           MessageBoxButtons.OK, MessageBoxIcon.Information);
       else
           MessageBox.Show("A is different from B.",
                       "Algebra of Sets",
                       MessageBoxButtons.OK, MessageBoxIcon.Information);
   
       if( A.IsSubsetOf(B))
           MessageBox.Show("A is a sub-set of B.",
                           "Algebra of Sets",
                           MessageBoxButtons.OK, MessageBoxIcon.Information);
       else
           MessageBox.Show("A is not a sub-set of B.",
                       "Algebra of Sets",
                       MessageBoxButtons.OK, MessageBoxIcon.Information);
   }

7. To execute, press Ctrl + F5
8. Add a few numbers to the left list box
9. Add the same numbers to the right list box in any order of your choice
10. Add a few more numbers to the right list. Here is an example:
     
    
11. Click Compare
     
    
12. Click OK
13. Close the form and return to your programming environment

A set is a subset of itself. Also, when two sets have the exact same members, each set is a subset of the other:

On the other hand, if you have a set A that is strictly a subset of another set B, this means there is at least one element in set B that is not a member of set A. In this case, we say that set A is a property subset of set B. This can be illustrated as follows:

To let you find out if a set is a proper subset of another set, the HashSet class is equipped with the IsProperSubsetOf() method. Its syntax is:

public bool IsProperSubsetOf(IEnumerable<T> other);

This method takes a HashSet value as arguments and performs a comparison on their elements:

• If both sets are the same, the method returns false
• If the argument has at least one element that is not in the variable that called it, the method returns false
• If all elements of the argument's set are members of the variable that called it, the method returns true

Here are two examples of calling this method:

private void FormLoaded(object sender, EventArgs ea)
{
    HashSet<string> Persons = new HashSet<string>();
    Persons.Add("Kevin Walchild");
    Persons.Add("Frank Maguire");
    Persons.Add("Jane Overton");
    Persons.Add("Daniel Albertson");

    HashSet<string> Students = new HashSet<string>();
    Students.Add("Jane Overton");
    Students.Add("Kevin Walchild");
    Students.Add("Daniel Albertson");
    Students.Add("Frank Maguire");

    HashSet<string> People = new HashSet<string>();
    People.Add("Raul Fitzgerald");
    People.Add("Kevin Walchild");
    People.Add("Tiffany Morales");
    People.Add("Daniel Albertson");
    People.Add("Michael Rhodia");
    People.Add("Frank Maguire");
    People.Add("Ruth Oyawale");
    People.Add("Jane Overton");
    People.Add("Paul Sullivan");

    if (Persons.IsProperSubsetOf(Students))
        MessageBox.Show("The Persons set is a proper subset of the Students set.",
                        "Collection Class",
                        MessageBoxButtons.OK, MessageBoxIcon.Information);
    else
       MessageBox.Show("The Persons set is not a proper subset of the Students set.",
                        "Collection Class",
                        MessageBoxButtons.OK, MessageBoxIcon.Information);

    if (Persons.IsProperSubsetOf(People))
        MessageBox.Show("The Persons set is a proper subset of the People set.",
                        "Collection Class",
                        MessageBoxButtons.OK, MessageBoxIcon.Information);
    else
        MessageBox.Show("The Persons set is not a proper subset of the People set.",
                        "Collection Class",
                        MessageBoxButtons.OK, MessageBoxIcon.Information);
}

1. Change the code of the Compare button as follows:

   private void btnCompare_Click(object sender, EventArgs e)
   {
       if (A.SetEquals(B))
           MessageBox.Show("A is equal to B.",
                           "Algebra of Sets",
                           MessageBoxButtons.OK, MessageBoxIcon.Information);
       else
           MessageBox.Show("A is different from B.",
                       	"Algebra of Sets",
                       	MessageBoxButtons.OK, MessageBoxIcon.Information);
   
       if( A.IsSubsetOf(B))
           MessageBox.Show("A is a sub-set of B.",
                           "Algebra of Sets",
                           MessageBoxButtons.OK, MessageBoxIcon.Information);
       else
           MessageBox.Show("A is not a sub-set of B.",
                       	"Algebra of Sets",
                       	MessageBoxButtons.OK, MessageBoxIcon.Information);
   
       if( A.IsProperSubsetOf(B))
           MessageBox.Show("A is a proper sub-set of B.",
                           "Algebra of Sets",
                           MessageBoxButtons.OK, MessageBoxIcon.Information);
       else
           MessageBox.Show("A is not a proper sub-set of B.",
                           "Algebra of Sets",
                           MessageBoxButtons.OK, MessageBoxIcon.Information);
   }

2. To execute, press Ctrl + F5
3. Add a few numbers to the left list box
4. Add the same numbers for the right list box in any order of your choice
    
   
5. Click Compare
    
   
6. Click OK
7. Add a few more numbers to the right list
8. Click Compare. Here is an example:
    
   
9. Click OK
10. Close the form and return to your programming environment