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Fix set theory formatting issues
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@ -29,11 +29,13 @@ These operators don't require a lot of text to describe.
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* `Z`, the set of all integers. `{…,-2,-1,0,1,2,…}`
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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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### The empty set
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* The set containing no items is called the empty set. Representation: `∅`
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* The empty set can be described as `∅ = {x|x ≠ x}`
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* The empty set is always unique.
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* The empty set is the subset of all sets.
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```
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A = {x|x∈N,x < 0}
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A = ∅
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@ -42,6 +44,7 @@ A = ∅
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|∅| = 0
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|{∅}| = 1
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```
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## Representing sets
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### Enumeration
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* List all items of the set, e.g. `A = {a,b,c,d}`
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@ -49,6 +52,7 @@ A = ∅
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### Description
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* Describes the features of all items in the set. Syntax: `{body|condtion}`
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```
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A = {x|x is a vowel}
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B = {x|x ∈ N, x < 10l}
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@ -84,6 +88,7 @@ C = {2x|x ∈ N}
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* The number of items in a set is called the base number of that set. Representation: `|A|`
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* If the base number of the set is finite, this set is a finite set.
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* If the base number of the set is infinite, this set is an infinite set.
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```
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A = {A,B,C}
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|A| = 3
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@ -94,6 +99,7 @@ B = {a,{b,c}}
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### Powerset
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* Let `A` be any set. The set that contains all possible subsets of `A` is called a powerset (written as `P(A)`).
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```
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P(A) = {x|x ⊆ A}
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@ -103,41 +109,54 @@ P(A) = {x|x ⊆ A}
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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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## "Generalized" operations
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### General union
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Better known as "flattening" of a set of sets.
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```
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∪A = {x|X∈A,x∈X}
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∪A={a,b,c,d,e,f}
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∪B={a}
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∪C=a∪{c,d}
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```
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### General intersection
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```
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∩ A = A1 ∩ A2 ∩ … ∩ An
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```
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