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discrete-math/01-sets-basics.md

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+# 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]]