Set Operations

When working with sets, we can perform various operations to combine, differentiate, or compare them.

For instance, you can merge the elements of two sets into a single set, or extract only the elements they have in common. The first is called the union, while the second is the intersection.

These operations are fundamental not just in pure mathematics but also in computer science, social sciences, and even philosophy.

Let’s break them down and see how they work.

Union (∪)

The union of two sets \( A \) and \( B \), written as \( A \cup B \), is the set that includes all elements that belong to at least one of the two sets.

For example, consider these two sets:

$$ A = \{1, 2, 3\} $$

$$ B = \{3, 4, 5\} $$

Their union consists of all elements from both sets:

\[ A \cup B = \{1, 2, 3, 4, 5\} \]

Notice that element \( 3 \), which appears in both sets, is listed only once since sets do not allow duplicate elements.

Intersection (∩)

The intersection of two sets \( A \) and \( B \), denoted as \( A \cap B \), is the set of elements that are common to both.

Using the same sets as before:

$$ A = \{1, 2, 3\} $$

$$ B = \{3, 4, 5\} $$

The intersection contains only the elements that appear in both sets:

\[ A \cap B = \{3\} \]

Here, the only common element is \( 3 \).

If two sets share no elements, their intersection is empty, represented by the empty set symbol \( \emptyset \).

For example:

$$ A = \{1, 2, 3\} $$

$$ C = \{6, 7\} $$

Since these two sets have no overlap, their intersection is:

\[ A \cap C = \emptyset \]

Set Difference (−)

The difference between two sets identifies elements that belong to one set but not the other. It is written as \( A - B \).

Consider these sets:

$$ A = \{1, 2, 3\} $$

$$ B = \{3, 4, 5\} $$

The difference \( A - B \) consists of the elements in \( A \) that are not in \( B \):

\[ A - B = \{1, 2\} \]

Likewise, the difference \( B - A \) consists of the elements in \( B \) that are not in \( A \):

\[ B - A = \{4, 5\} \]

In both cases, element \( 3 \) is omitted because it belongs to both sets.

It’s important to note that set difference is not commutative—swapping the sets gives a different result.

Complement (\( A^C \))

The complement of a set \( A \), written as \( A^C \), consists of all elements that are not in \( A \), relative to a given universal set \( U \).

For example, if the universal set is:

$$ U = \{1, 2, 3, 4, 5, 6, 7\} $$

And we define the set \( A \) as:

$$ A = \{2, 4, 6\} $$

The complement of \( A \) relative to \( U \) is:

\[ A^C = \{1, 3, 5, 7\} \]

In other words, the complement consists of all elements in \( U \) that are not in \( A \) (i.e., everything except 2, 4, and 6).

Symmetric Difference (Δ) or Exclusive Set

The symmetric difference of two sets \( A \) and \( B \), denoted as \( A \triangle B \), consists of elements that belong to either \( A \) or \( B \), but not both.

For instance, consider:

$$ A = \{1, 2, 3\} $$

$$ B = \{3, 4, 5\} $$

The symmetric difference is:

$$ A \triangle B = \{1, 2, 4, 5\} $$

Element \( 3 \) is excluded because it appears in both sets.

Unlike set difference, the symmetric difference is commutative—reversing the sets does not affect the outcome.

It can also be expressed as the union of two differences:

$$ A \triangle B = (A - B) \cup (B - A) $$

Why Are Set Operations Important?

Set operations aren’t just abstract mathematical concepts—they have real-world applications across multiple fields.

In computer science, they are used in databases to filter and compare data sets. In statistics, they help analyze population segments and study overlapping categories. In logic and philosophy, they provide a structured way to classify concepts and construct arguments.

Fundamental Properties of Set Operations

Now that we've explored set operations, it's crucial to understand the fundamental properties that govern them.

Some of these properties mirror those found in arithmetic, while others are unique to set theory.

Property	Expression
Idempotent Property	\( A \cap A = A \) \( A \cup A = A \)
Commutative Property of Intersection	\( A \cap B = B \cap A \)
Commutative Property of Union	\( A \cup B = B \cup A \)
Associative Property of Intersection	\( A \cap (B \cap C) = (A \cap B) \cap C \)
Associative Property of Union	\( A \cup (B \cup C) = (A \cup B) \cup C \)
Absorption Laws	\( A \cap (A \cup B) = A \) \( A \cup (A \cap B) = A \)
Distributive Property of Intersection over Union	\( A \cap (B \cup C) = (A \cap B) \cup (A \cap C) \)
Distributive Property of Union over Intersection	\( A \cup (B \cap C) = (A \cup B) \cap (A \cup C) \)
De Morgan's Laws	\( \overline{A \cap B} = \overline{A} \cup \overline{B} \) \( \overline{A \cup B} = \overline{A} \cap \overline{B} \)

In the next section, we'll take a deeper dive into these properties, illustrating each with clear, concrete examples.