# Sets: Basics and Operations

**Source:** MIT 6.1200J Lecture 01

## What is a Set?
A **set** is a collection of objects.

### Properties
- **No duplicates:** {1, 2, 2, 3} = {1, 2, 3}
- **Order doesn't matter:** {1, 2, 3} = {3, 1, 2}
- **Can be infinite:** ℕ = {0, 1, 2, ...}
- **Can contain other sets:** B = {2, {3, 4}, ∅}

## Basic Notation

### Membership
- **∈** (element of): 6 ∈ A means "6 is in set A"
- **∉** (not element of): {1, 2} ∉ A

### Subset
- **⊆** (subset): S ⊆ T means all elements of S are also in T
- Example: {1, 2} ⊆ {0, 1, 2, 6}

## Common Sets
- **ℕ** (Natural numbers): {0, 1, 2, ...}
- **ℤ** (Integers): {..., -2, -1, 0, 1, 2, ...}
- **∅** or **{}** (Empty set): The set with no elements

## Set-Builder Notation
Used to define sets with predicates:

```
{n ∈ ℕ | isPrime(n)} = {2, 3, 5, 7, 11, ...}
```

Reads as: "The set of all n in ℕ such that isPrime(n) is true"

Can also use colon instead of vertical bar: `{n ∈ ℕ : isPrime(n)}`

## Set Operations

### Intersection (∩)
A ∩ B = {elements in both A and B}

### Union (∪)
A ∪ B = {elements in A or B (or both)}

### Difference (\ or −)
A \ B = {elements in A that are not in B}

## Ordered Tuples
When **order matters**, use parentheses (not braces):

- **(6, 1, 2, 0) ≠ (2, 1, 6, 0)**
- Duplicates allowed: (1, 2, 2, 3) is valid
- Set operations (∩, ∪, etc.) don't apply to tuples

## Related
- [[../logic-proofs/00-index|Logic & Proofs]]
- [[00-index|Discrete Math Index]]