# Intersection of Sets — Symbol, Definition, Examples

TL;DR

The intersection of sets A and B, written A∩B, is the set of elements that are in both A and B. This article covers the formal definition, the symbol, properties, three worked examples, the common slips, and a side-by-side comparison with union, set difference, and complement.

## The Overlap Between Two Collections

Imagine two clubs at a school — the Chess Club and the Robotics Club. Some students belong to one. Some belong to the other. A few belong to both. The "both" group is the **intersection of sets** — the overlap.

The intersection picks out exactly the elements that appear in every set being intersected. Symbolically:

A∩B={x:x∈A and x∈B}.

If no elements are shared, A∩B=∅ (the empty set), and the sets are called **disjoint**.

## The Symbol and Formal Definition

The intersection symbol is ∩ — a downward-pointing arc that looks like an upside-down U. It can be read as "intersected with" or simply "and."

x∈A∩B⟺x∈A and x∈B.

The intersection extends to any number of sets: A1∩A2∩⋯∩An is the set of elements that belong to _every_ Ai.

### Properties at a glance

| Property | Statement | Meaning |
| --- | --- | --- |
| **Commutative** | A∩B=B∩A | Order doesn't matter |
| **Associative** | (A∩B)∩C=A∩(B∩C) | Grouping doesn't matter |
| **Identity** | A∩U=A | Intersecting with the universal set U returns the original set |
| **Domination** | A∩∅=∅ | Intersection with empty set is empty |
| **Idempotent** | A∩A=A | Intersecting a set with itself returns itself |
| **Distributive** | A∩(B∪C)=(A∩B)∪(A∩C) | Intersection distributes over union |
| **De Morgan** | (A∩B)'=A'∪B' | Complement of intersection equals union of complements |

## Intersection vs Other Set Operations — Side by Side

| Operation | Symbol | What it contains | Example: A={1,2,3}, B={2,3,4} |
| --- | --- | --- | --- |
| **Union** | A∪B | Elements in A _or_ B (or both) | {1, 2, 3, 4} |
| **Intersection** | A∩B | Elements in _both_ A and B | {2, 3} |
| **Difference** | A−B | Elements in A but _not_ in B | {1} |
| **Symmetric difference** | A△B | Elements in A or B but _not both_ | {1, 4} |
| **Complement** | A' | Elements in the universe U but _not_ in A | Depends on U |
| **Cartesian product** | A×B | All ordered pairs (a,b) | {(1,2), (1,3), (1,4), (2,2),…} |

The pattern: union, intersection, and difference are the three "basic" Boolean operations; complement is the unary one; symmetric difference and Cartesian product are derived.

### The cardinality formula

For finite sets,

|A∪B|=|A|+|B|−|A∩B|.

This is the **inclusion-exclusion principle** for two sets. Rearranged:

|A∩B|=|A|+|B|−|A∪B|.

The principle generalises to three or more sets — but with alternating-sign terms for triple intersections, quadruple, and so on. Almost every Grade 11 board question on sets uses inclusion-exclusion for two or three sets.

## Three Worked Examples — Quick, Standard, Stretch

### **Quick.** Find A∩B for A={1,2,3,4,5} and B={4,5,6,7,8}.

Scan for elements in both. 4 is in both. 5 is in both. Everything else is in one set only.

A∩B={4, 5}.  
**Final answer:** A∩B={4, 5}.

### **Standard.** Out of 100 students in a class, 60 like maths and 45 like science; 25 like both. How many like neither?

Let M = set of maths-likers (|M|=60), S = set of science-likers (|S|=45), |M∩S|=25.

The number who like _at least one_ is:

|M∪S|=|M|+|S|−|M∩S|=60+45−25=80.

The number who like _neither_ is 100−|M∪S|=100−80=20.

**Final answer:** 20 students like neither.

### **Stretch.** Find A∩B∩C for A={x∈N:x≤20}, B={x:x is a multiple of 3}, C={x:x is a multiple of 4}.

The intersection requires elements in _all three_ sets. A number that is a multiple of both 3 and 4 is a multiple of their LCM, which is 12. So we want multiples of 12 that are ≤20. Just 12.

**Final answer:** A∩B∩C={12}.

## Why Intersection Matters

Intersection is one of the three foundational set operations — without it, set theory is just collections.

- **Probability.** The probability of two events both happening is the probability of the **intersection** of the events: P(A∩B). Independence is the property P(A∩B)=P(A)⋅P(B).

- **Database queries.** Every SQL `INNER JOIN` is an intersection — return only rows where the matching key exists in both tables.

- **Logic and computer science.** Boolean AND is intersection. A program that returns "true if the user is logged in AND has admin privileges" is computing membership in the intersection of two sets.

- **Statistics.** Confidence intervals from two independent samples are tightened by intersecting them — the result is the set of parameter values consistent with both samples.

- **Geometry.** Two lines meet at a single point (their intersection); a line and a circle meet in two points (or one tangent, or none); planes intersect in lines or points.

## Where to Watch Your Step on Intersection

### **Mistake 1: Confusing ∩ with ∪.**

A student reads A∩B and writes the union — all elements that appear in either set. **Don't do this:** Swap the symbols.

### **Mistake 2: Listing duplicate elements.**

A student computes {1,2,2,3}∩{2,3,4} and writes {2,2,3}. **Don't do this:** Carry duplicates into the result.

### **Mistake 3: Forgetting that the intersection can be empty.**

A student computes {1,3,5}∩{2,4,6}, finds no common elements, and writes "no answer" or leaves the question blank. **Don't do this:** The intersection of disjoint sets is the **empty set**, written ∅ or {}.

## Conclusion

- The intersection of sets A and B, written A∩B, is the set of elements in both.
- The intersection symbol ∩ reads as "intersection" or "and"; the dual symbol ∪ is union (or).
- The cardinality formula |A∪B|=|A|+|B|−|A∩B| is the workhorse of Grade 11 set problems.
- Intersection of disjoint sets is ∅ — the empty set is a valid answer.
- Intersection generalises to any number of sets and is the foundation of Boolean AND, SQL inner joins, probability of joint events, and geometric meeting points.

## Try Intersection Yourself — Three Problems

1. Find A∩B for A={2,4,6,8,10} and B={1,4,9,16,25}.
2. In a survey of 80 households, 50 own a car and 35 own a bicycle; 20 own both. How many own neither?
3. Given A={x:1≤x≤10} and B={x:x is prime}, find A∩B.