61 lines
1.5 KiB
Markdown
61 lines
1.5 KiB
Markdown
# Sets: Basics and Operations
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**Source:** MIT 6.1200J Lecture 01
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## What is a Set?
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A **set** is a collection of objects.
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### Properties
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- **No duplicates:** {1, 2, 2, 3} = {1, 2, 3}
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- **Order doesn't matter:** {1, 2, 3} = {3, 1, 2}
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- **Can be infinite:** ℕ = {0, 1, 2, ...}
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- **Can contain other sets:** B = {2, {3, 4}, ∅}
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## Basic Notation
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### Membership
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- **∈** (element of): 6 ∈ A means "6 is in set A"
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- **∉** (not element of): {1, 2} ∉ A
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### Subset
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- **⊆** (subset): S ⊆ T means all elements of S are also in T
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- Example: {1, 2} ⊆ {0, 1, 2, 6}
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## Common Sets
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- **ℕ** (Natural numbers): {0, 1, 2, ...}
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- **ℤ** (Integers): {..., -2, -1, 0, 1, 2, ...}
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- **∅** or **{}** (Empty set): The set with no elements
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## Set-Builder Notation
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Used to define sets with predicates:
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```
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{n ∈ ℕ | isPrime(n)} = {2, 3, 5, 7, 11, ...}
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```
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Reads as: "The set of all n in ℕ such that isPrime(n) is true"
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Can also use colon instead of vertical bar: `{n ∈ ℕ : isPrime(n)}`
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## Set Operations
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### Intersection (∩)
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A ∩ B = {elements in both A and B}
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### Union (∪)
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A ∪ B = {elements in A or B (or both)}
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### Difference (\ or −)
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A \ B = {elements in A that are not in B}
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## Ordered Tuples
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When **order matters**, use parentheses (not braces):
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- **(6, 1, 2, 0) ≠ (2, 1, 6, 0)**
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- Duplicates allowed: (1, 2, 2, 3) is valid
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- Set operations (∩, ∪, etc.) don't apply to tuples
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## Related
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- [[../logic-proofs/00-index|Logic & Proofs]]
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- [[00-index|Discrete Math Index]]
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