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133 lines
4.8 KiB
Markdown
133 lines
4.8 KiB
Markdown
---
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category: Algorithms & Data Structures
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name: Set theory
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contributors:
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---
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Set theory is a branch of mathematics that studies sets, their operations, and their properties.
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* A set is a collection of disjoint items.
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## Basic symbols
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### Operators
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* the union operator, `∪`, pronounced "cup", means "or";
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* the intersection operator, `∩`, pronounced "cap", means "and";
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* the exclusion operator, `\`, means "without";
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* the complement operator, `'`, means "the inverse of";
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* the cross operator, `×`, means "the Cartesian product of".
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### Qualifiers
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* the colon qualifier, `:`, means "such that";
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* the membership qualifier, `∈`, means "belongs to";
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* the subset qualifier, `⊆`, means "is a subset of";
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* the proper subset qualifier, `⊂`, means "is a subset of but is not equal to".
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### Canonical sets
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* `∅`, the empty set, i.e. the set containing no items;
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* `ℕ`, the set of all natural numbers;
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* `ℤ`, the set of all integers;
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* `ℚ`, the set of all rational numbers;
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* `ℝ`, the set of all real numbers.
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There are a few caveats to mention regarding the canonical sets:
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1. Even though the empty set contains no items, the empty set is a subset of itself (and indeed every other set);
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2. Mathematicians generally do not universally agree on whether zero is a natural number, and textbooks will typically explicitly state whether or not the author considers zero to be a natural number.
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### Cardinality
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The cardinality, or size, of a set is determined by the number of items in the set. The cardinality operator is given by a double pipe, `|...|`.
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For example, if `S = { 1, 2, 4 }`, then `|S| = 3`.
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### The Empty Set
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* The empty set can be constructed in set builder notation using impossible conditions, e.g. `∅ = { x : x ≠ x }`, or `∅ = { x : x ∈ N, x < 0 }`;
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* the empty set is always unique (i.e. there is one and only one empty set);
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* the empty set is a subset of all sets;
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* the cardinality of the empty set is 0, i.e. `|∅| = 0`.
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## Representing sets
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### Literal Sets
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A set can be constructed literally by supplying a complete list of objects contained in the set. For example, `S = { a, b, c, d }`.
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Long lists may be shortened with ellipses as long as the context is clear. For example, `E = { 2, 4, 6, 8, ... }` is clearly the set of all even numbers, containing an infinite number of objects, even though we've only explicitly written four of them.
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### Set Builder
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Set builder notation is a more descriptive way of constructing a set. It relies on a _subject_ and a _predicate_ such that `S = { subject : predicate }`. For example,
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```
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A = { x : x is a vowel } = { a, e, i, o, u, y}
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B = { x : x ∈ N, x < 10 } = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 }
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C = { x : x = 2k, k ∈ N } = { 0, 2, 4, 6, 8, ... }
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```
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Sometimes the predicate may "leak" into the subject, e.g.
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```
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D = { 2x : x ∈ N } = { 0, 2, 4, 6, 8, ... }
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```
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## Relations
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### Membership
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* If the value `a` is contained in the set `A`, then we say `a` belongs to `A` and represent this symbolically as `a ∈ A`.
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* If the value `a` is not contained in the set `A`, then we say `a` does not belong to `A` and represent this symbolically as `a ∉ A`.
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### Equality
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* If two sets contain the same items then we say the sets are equal, e.g. `A = B`.
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* Order does not matter when determining set equality, e.g. `{ 1, 2, 3, 4 } = { 2, 3, 1, 4 }`.
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* Sets are disjoint, meaning elements cannot be repeated, e.g. `{ 1, 2, 2, 3, 4, 3, 4, 2 } = { 1, 2, 3, 4 }`.
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* Two sets `A` and `B` are equal if and only if `A ⊆ B` and `B ⊆ A`.
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## Special Sets
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### The Power Set
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* Let `A` be any set. The set that contains all possible subsets of `A` is called a "power set" and is written as `P(A)`. If the set `A` contains `n` elements, then `P(A)` contains `2^n` elements.
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```
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P(A) = { x : x ⊆ A }
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```
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## Set operations among two sets
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### Union
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Given two sets `A` and `B`, the union of the two sets are the items that appear in either `A` or `B`, written as `A ∪ B`.
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```
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A ∪ B = { x : x ∈ A ∪ x ∈ B }
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```
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### Intersection
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Given two sets `A` and `B`, the intersection of the two sets are the items that appear in both `A` and `B`, written as `A ∩ B`.
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```
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A ∩ B = { x : x ∈ A, x ∈ B }
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```
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### Difference
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Given two sets `A` and `B`, the set difference of `A` with `B` is every item in `A` that does not belong to `B`.
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```
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A \ B = { x : x ∈ A, x ∉ B }
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```
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### Symmetrical difference
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Given two sets `A` and `B`, the symmetrical difference is all items among `A` and `B` that doesn't appear in their intersections.
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```
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A △ B = { x : ((x ∈ A) ∩ (x ∉ B)) ∪ ((x ∈ B) ∩ (x ∉ A)) }
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A △ B = (A \ B) ∪ (B \ A)
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```
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### Cartesian product
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Given two sets `A` and `B`, the cartesian product between `A` and `B` consists of a set containing all combinations of items of `A` and `B`.
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```
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A × B = { (x, y) | x ∈ A, y ∈ B }
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```
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