Key Properties of Set Operations
Set operations follow a well-defined set of logical properties. Some of these mirror familiar arithmetic properties, while others are unique to set theory. Below are the most important ones:
| 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 | \( \overline{A \cap B} = \overline{A} \cup \overline{B} \) \( \overline{A \cup B} = \overline{A} \cap \overline{B} \) |
Now, let’s take a closer look at the most significant properties, with clear examples to illustrate them.
Idempotent Property
This property states that taking the union or intersection of a set with itself yields the same set—it remains unchanged.
\[ A \cap A = A \]
\[ A \cup A = A \]
For example, if \( A = \{1, 2, 3\} \), then:
\( A \cap A = \{1, 2, 3\} \), since intersecting a set with itself does not remove any elements.
\( A \cup A = \{1, 2, 3\} \), since union with itself does not introduce anything new.
Commutative Property
The order of sets does not affect the result of union or intersection.
\[ A \cap B = B \cap A \]
\[ A \cup B = B \cup A \]
For example, if \( A = \{1, 2\} \) and \( B = \{3, 4\} \), then:
\( A \cap B = B \cap A = \emptyset \) (the empty set).
\( A \cup B = B \cup A = \{1, 2, 3, 4\} \).
Associative Property
Grouping does not affect the result when performing unions or intersections of multiple sets.
\[ A \cap (B \cap C) = (A \cap B) \cap C \]
\[ A \cup (B \cup C) = (A \cup B) \cup C \]
For instance, if \( A = \{1, 2\} \), \( B = \{2, 3\} \), and \( C = \{3, 4\} \), then:
\( A \cap (B \cap C) = (A \cap B) \cap C = \emptyset \).
\( A \cup (B \cup C) = (A \cup B) \cup C = \{1, 2, 3, 4\} \).
Absorption Laws
These laws describe how a set interacts with unions and intersections involving itself and another set.
\[ A \cap (A \cup B) = A \]
\[ A \cup (A \cap B) = A \]
For example, if \( A = \{1, 2\} \) and \( B = \{2, 3\} \), then:
\( A \cap (A \cup B) = A \), since \( A \cup B = \{1, 2, 3\} \) and intersecting it with \( A \) leaves \( \{1, 2\} \).
\( A \cup (A \cap B) = A \), since \( A \cap B = \{2\} \) and taking the union with \( A \) does not add anything new.
Distributive Property
Union and intersection distribute over each other, much like multiplication and addition do in arithmetic.
\[ A \cap (B \cup C) = (A \cap B) \cup (A \cap C) \]
\[ A \cup (B \cap C) = (A \cup B) \cap (A \cup C) \]
For example, if \( A = \{1, 2\} \), \( B = \{2, 3\} \), and \( C = \{3, 4\} \), then:
\( A \cap (B \cup C) = \{1, 2\} \cap \{2, 3, 4\} = \{2\} \).
\( (A \cap B) \cup (A \cap C) = \{2\} \cup \emptyset = \{2\} \).
De Morgan’s Laws
These fundamental rules describe how complements interact with unions and intersections.
\[ \overline{A \cap B} = \overline{A} \cup \overline{B} \]
\[ \overline{A \cup B} = \overline{A} \cap \overline{B} \]
The overline denotes the complement of a set relative to the universal set.
For example, if the universal set is \( U = \{1, 2, 3, 4, 5\} \) and the sets are \( A = \{1, 2\} \) and \( B = \{2, 3\} \), then:
\( \overline{A} = \{3, 4, 5\} \) and \( \overline{B} = \{1, 4, 5\} \).
\( \overline{A \cap B} = \overline{\{2\}} = \{1, 3, 4, 5\} \), which equals \( \overline{A} \cup \overline{B} \).
\( \overline{A \cup B} = \overline{\{1, 2, 3\}} = \{4, 5\} \), which equals \( \overline{A} \cap \overline{B} \).
Understanding these properties is essential for mastering set theory and its applications in logic, mathematics, and beyond.
