In this section, we will discuss properties specific to sets that were introduced in the previous section. When performing operations on sets, we use terms such as Intersection, Union, and Complement. These properties are collectively known as set operations. Let us now explore these operations in detail below.

1.1.1. Intersetion

In general terms, the intersection of sets refers to a new set formed by the common elements of two or more sets. The intersection operation is usually denoted by the symbol ∩. For example, given two sets ( A ) and ( B ):

• It is represented as ( A ∩ B ).
• If ( A ∩ B = ∅ ), then ( A ) and ( B ) do not share any common elements and are said to be disjoint.

1.1.2. Union

In general terms, the union of two or more sets is a new set that contains all the elements from the sets involved. The union operation is typically denoted by the symbol ∪. In a union, all elements from each set are combined, and any repeated elements are listed only once.

• It is represented as A ∪ B.
• If x ∈ A ∪ B, then x is found in either A, B, or both. Technically, this operation is called inclusive-or because it includes the possibility of x being in both sets.

1.1.3. Complement

The complement of a set is a new set consisting of the elements in the universal set that are not in the given set. In other words, the complement of a set includes the elements of the universal set that are not present in the set itself. The complement operation is usually represented as A′ or A^c.

Let us illustrate this with an example:

E = {1, 2, 3, 4, 5} and A = {2, 4}

A′ = {1, 3, 5}

The complement operation allows for the identification of missing elements within sets and the examination of elements outside the given set. Mathematically, this can be expressed as:

A′ = {x : x ∈ E and x ∉ A}

Now, let us discuss how two sets can be considered equal. This is only the case when they have exactly the same elements. A special condition arises when comparing sets.

For example: For example, if all elements of set ( A ) are also elements of set ( B );

a. Sets ( A ) and ( B ) may be the same set, or
b. It can be stated that ( A ) is a proper subset of ( B ), denoted as ( A ⊂ B ).
c. To express the possibility that ( A ) and ( B ) have exactly the same elements, we use ( A ⊆ B ).
d. If set ( A ) contains all the elements of set ( B ) and ( A \neq B ), it is stated that ( A ) is a proper superset of ( B ), denoted as ( A ⊃ B ).
e. If we accept the possibility that ( A ) and ( B ) are the same set, we write ( A ⊇ B ).
f. The empty set ({}) is a subset of any set, denoted by ({} ⊂ A).

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Now, let us illustrate each of these cases with examples. This will greatly enhance understanding.

For example:

a. If ( A = {1, 2, 3} ) and ( B = {1, 2, 3} ), then we can say A = B.

b. If ( A = {1, 2} ) and ( B = {1, 2, 3} ),

then ( A ⊂ B ). All elements of ( A ) are in ( B ), but ( A ≠ B ), making ( A ) a proper subset.

c. If ( A = {1, 2, 3} ) and ( B = {1, 2, 3} ), then ( A ⊆ B ).

This represents the more general case of ( A ) being a subset of ( B ), including the possibility that ( A ) is equal to ( B ).

d. If ( D = {1, 2, 3, 4} ) and ( E = {1, 2} ), then ( D ⊃ E ).

( D ) is a proper superset of ( E ) as it contains all elements of ( E ) and more.

e. If ( F = {1, 2, 3} ) and ( B = {1, 2, 3} ), then ( F ⊇ B ).

Here, ( F ) and ( B ) have the same elements, so ( F ⊇ B ) and ( B ⊇ F ).