Intersection of Sets

The intersection of two sets A and B is the set of all elements that the two sets have in common.

The symbol used to denote an intersection is ∩.

The expression A∩B is read as "the intersection of A and B" or simply "A intersect B".

Intersection of a Family of Sets. When working with more than two sets, mathematicians often use a compact notation to represent their common intersection: $$ \bigcap_{i \in I} A_i = \{ x \mid x \in A_i \text{ for every } i \in I \} $$ This notation denotes the set of all elements that belong to every set in the family.

If two sets have no elements in common, their intersection is the empty set.

In this situation, the sets are called disjoint sets.

A Practical Example

Consider the following finite sets:

$$ A = \{ 2,5,6,7,8 \} $$

$$ B = \{ 1,3,4,6,7,9 \} $$

To find their intersection, identify the elements that appear in both sets.

$$ A = \{ 2,5,\color{red}6,\color{red}7,8 \} $$

$$ B = \{ 1,3,4,\color{red}6,\color{red}7,9 \} $$

The common elements are 6 and 7. Therefore:

$$ A \cap B = \{ 6,7 \} $$

A Venn diagram provides a visual representation of the intersection:

The overlapping region contains exactly the elements that belong to both sets.

Types of Intersection

The result of an intersection depends on the relationship between the sets involved.

• The Intersection Is a Proper Subset of Both Sets

  Example. Let $$ A = \{ 2,5,6,7,8 \} $$ $$ B = \{ 1,3,4,6,7,9 \} $$ Then $$ A \cap B = \{ 6,7 \} $$ Since the intersection contains only some of the elements of each set: $$ A \cap B \subset A $$ $$ A \cap B \subset B $$
  

• The Intersection Coincides with One of the Sets

  If one set is contained entirely within another, the intersection is equal to the smaller set.

  Example. Let $$ A = \{ 3,6,7 \} $$ $$ B = \{ 1,3,4,6,7,9 \} $$ Then $$ A \cap B = \{ 3,6,7 \} $$ which is exactly the set A.
  
  Therefore: $$ A \cap B = A $$ and $$ A \cap B \subset B $$

• The Intersection Is the Empty Set

  This occurs when the sets are disjoint and share no common elements.

  Example. Let $$ A = \{ 2,5,8 \} $$ $$ B = \{ 1,3,4,6,7,9 \} $$ Then $$ A \cap B = \varnothing $$
  
  Since the empty set is a subset of every set: $$ A \cap B \subseteq A $$ $$ A \cap B \subseteq B $$

• The Sets Are Equal

  If two sets contain exactly the same elements, their intersection is identical to both sets.

  Example. Let $$ A = \{ 1,2,3,4 \} $$ $$ B = \{ 1,2,3,4 \} $$ Then $$ A \cap B = A = B $$
  
  In this case, every element belongs to both sets, so: $$ A \subseteq B $$ $$ B \subseteq A $$

Properties of Intersection

Like many operations in mathematics, intersection satisfies several important algebraic properties.

• Commutative Property
  The order of the sets does not matter: $$ A \cap B = B \cap A $$

  

• Associative Property
  The way the sets are grouped does not affect the result: $$ (A \cap B) \cap C = A \cap (B \cap C) $$

  

• Distributive Property over Union
  Intersecting a set with a union produces the same result as taking the union of the individual intersections: $$ A \cap (B \cup C) = (A \cap B) \cup (A \cap C) $$

  

These properties play a fundamental role in set theory, logic, probability, and many other areas of mathematics.