The intersection of two or more sets is the set of elements that are common to all of the sets. It is denoted by the symbol ∩. If sets A and B share elements, their intersection, A ∩ B, is a new set containing only those shared elements.
Set intersection represents the logical 'AND' operation. An element is included in the intersection if and only if it belongs to the first set AND the second set AND all other sets involved.
| Symbol | Meaning |
|---|---|
| A ∩ B | The intersection of sets A and B. |
| x ∈ A | The element x is a member of set A. |
| ∅ | The empty set, which contains no elements. |
| U | The universal set, which contains all possible elements under consideration. |
| |A| | The cardinality of set A, meaning the number of elements in the set. |
An intersection is typically visualized using a Venn diagram. Imagine two overlapping circles, labeled 'A' and 'B'. Each circle represents a set. The region where the two circles overlap contains the elements that are in both set A and set B. This overlapping area is the graphical representation of A ∩ B.
We can prove that A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) by showing that any element in the set on the left is also in the set on the right, and vice versa.
Part 1: Show that A ∩ (B ∪ C) ⊆ (A ∩ B) ∪ (A ∩ C)
Let x be an arbitrary element of A ∩ (B ∪ C). By the definition of intersection:
By the definition of union, if x ∈ (B ∪ C), then x ∈ B or x ∈ C. This gives two cases:
Case 1: If x ∈ B. Since we know x ∈ A, it follows that x ∈ (A ∩ B).
Case 2: If x ∈ C. Since we know x ∈ A, it follows that x ∈ (A ∩ C).
In either case, x belongs to (A ∩ B) or to (A ∩ C). Therefore, by the definition of union:
Part 2: Show that (A ∩ B) ∪ (A ∩ C) ⊆ A ∩ (B ∪ C)
The steps are reversible. Let y be an arbitrary element of (A ∩ B) ∪ (A ∩ C). By the definition of union, y ∈ (A ∩ B) or y ∈ (A ∩ C). This means (y ∈ A and y ∈ B) or (y ∈ A and y ∈ C). Factoring out the common term 'y ∈ A', we get y ∈ A and (y ∈ B or y ∈ C). This simplifies to y ∈ A and y ∈ (B ∪ C), which means y ∈ A ∩ (B ∪ C).
Since both sets are subsets of each other, they must be equal.
Database Queries: In SQL, the INTERSECT operator or an INNER JOIN is used to find records that exist in two or more tables or query results. This is a direct application of set intersection to filter data.
Market Research: Analysts find the intersection of different customer segments. For example, finding the set of customers who are both 'under 30' and 'have purchased a specific product' allows for highly targeted marketing campaigns.
Genetics and Medicine: Researchers look for the intersection of genes present in individuals with a specific disease to identify potential genetic markers. In diagnostics, a doctor finds the intersection of a patient's symptoms with the known symptoms of various illnesses.
Computer Science: In web search engines, a query with multiple words (e.g., 'red convertible car') returns pages that are in the intersection of the sets of pages containing 'red', 'convertible', and 'car'.
Finding Mutual Friends
On social media platforms, when you view another person's profile, the 'mutual friends' list is the intersection of your set of friends and their set of friends. It identifies the people you both know in common.
Recipe Planning
When deciding what to cook, you might mentally find the intersection of two sets: the ingredients you currently have in your pantry and the ingredients required for a specific recipe. If the intersection is the same as the recipe's ingredient set, you can make the dish without a trip to the store.
Cross-Referencing Documents
Lawyers and researchers often need to find common themes or keywords that appear in multiple documents. This process is an intersection of the sets of keywords from each document, helping to identify central topics or evidence.
The nature of an intersection is defined by the relationship between the sets involved. The result can be classified based on whether the sets share elements.
| Intersection Type | Description | Example |
|---|---|---|
| Non-empty Intersection | The sets have at least one element in common. The resulting intersection set contains these shared elements. | A={1,2}, B={2,3} → A∩B={2} |
| Empty Intersection (Disjoint Sets) | The sets have no elements in common. Their intersection is the empty set (∅). | A={1,2}, B={3,4} → A∩B=∅ |
| Intersection as a Subset | One set is a complete subset of the other. The intersection is equal to the smaller set. | A={1,2}, B={1,2,3} → A∩B={1,2}=A |
Confusing Intersection with Union: A very common error is to list all elements from both sets (union) instead of only the elements that appear in both sets (intersection). Remember, intersection means 'AND' (must be in both), while union means 'OR' (can be in either).
Incorrect Notation for Empty Intersection: When two sets have no common elements (they are disjoint), their intersection is the empty set, denoted as ∅ or {}. It is incorrect to write the intersection as {∅} (a set containing the empty set) or 0 (the number zero).
Forgetting Elements in Multiple Sets: When finding the intersection of three or more sets (A ∩ B ∩ C), an element must be present in ALL of the sets. It's easy to mistakenly include an element that is only in two of the three sets. Always check against every single set.