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Fix set theory formatting issues

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Ben Levy 2020-07-04 22:46:46 -04:00 committed by GitHub
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@ -29,11 +29,13 @@ These operators don't require a lot of text to describe.
* `Z`, the set of all integers. `{…,-2,-1,0,1,2,…}` * `Z`, the set of all integers. `{…,-2,-1,0,1,2,…}`
* `Q`, the set of all rational numbers. * `Q`, the set of all rational numbers.
* `R`, the set of all real numbers. * `R`, the set of all real numbers.
### The empty set ### The empty set
* The set containing no items is called the empty set. Representation: `∅` * The set containing no items is called the empty set. Representation: `∅`
* The empty set can be described as `∅ = {x|x ≠ x}` * The empty set can be described as `∅ = {x|x ≠ x}`
* The empty set is always unique. * The empty set is always unique.
* The empty set is the subset of all sets. * The empty set is the subset of all sets.
``` ```
A = {x|x∈N,x < 0} A = {x|x∈N,x < 0}
A = ∅ A = ∅
@ -42,6 +44,7 @@ A = ∅
|∅| = 0 |∅| = 0
|{∅}| = 1 |{∅}| = 1
``` ```
## Representing sets ## Representing sets
### Enumeration ### Enumeration
* List all items of the set, e.g. `A = {a,b,c,d}` * List all items of the set, e.g. `A = {a,b,c,d}`
@ -49,6 +52,7 @@ A = ∅
### Description ### Description
* Describes the features of all items in the set. Syntax: `{body|condtion}` * Describes the features of all items in the set. Syntax: `{body|condtion}`
``` ```
A = {x|x is a vowel} A = {x|x is a vowel}
B = {x|x ∈ N, x < 10l} B = {x|x ∈ N, x < 10l}
@ -84,6 +88,7 @@ C = {2x|x ∈ N}
* The number of items in a set is called the base number of that set. Representation: `|A|` * The number of items in a set is called the base number of that set. Representation: `|A|`
* If the base number of the set is finite, this set is a finite set. * If the base number of the set is finite, this set is a finite set.
* If the base number of the set is infinite, this set is an infinite set. * If the base number of the set is infinite, this set is an infinite set.
``` ```
A = {A,B,C} A = {A,B,C}
|A| = 3 |A| = 3
@ -94,6 +99,7 @@ B = {a,{b,c}}
### Powerset ### Powerset
* Let `A` be any set. The set that contains all possible subsets of `A` is called a powerset (written as `P(A)`). * Let `A` be any set. The set that contains all possible subsets of `A` is called a powerset (written as `P(A)`).
``` ```
P(A) = {x|x ⊆ A} P(A) = {x|x ⊆ A}
@ -103,41 +109,54 @@ P(A) = {x|x ⊆ A}
## Set operations among two sets ## Set operations among two sets
### Union ### Union
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`. 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`.
``` ```
A ∪ B = {x|x∈A∨x∈B} A ∪ B = {x|x∈A∨x∈B}
``` ```
### Intersection ### Intersection
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`. 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`.
``` ```
A ∩ B = {x|x∈A,x∈B} A ∩ B = {x|x∈A,x∈B}
``` ```
### Difference ### Difference
Given two sets `A` and `B`, the set difference of `A` with `B` is every item in `A` that does not belong to `B`. Given two sets `A` and `B`, the set difference of `A` with `B` is every item in `A` that does not belong to `B`.
``` ```
A \ B = {x|x∈A,x∉B} A \ B = {x|x∈A,x∉B}
``` ```
### Symmetrical difference ### Symmetrical difference
Given two sets `A` and `B`, the symmetrical difference is all items among `A` and `B` that doesn't appear in their intersections. Given two sets `A` and `B`, the symmetrical difference is all items among `A` and `B` that doesn't appear in their intersections.
``` ```
A △ B = {x|(x∈A∧x∉B)∨(x∈B∧x∉A)} A △ B = {x|(x∈A∧x∉B)∨(x∈B∧x∉A)}
A △ B = (A \ B) ∪ (B \ A) A △ B = (A \ B) ∪ (B \ A)
``` ```
### Cartesian product ### Cartesian product
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`. 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`.
``` ```
A × B = { {x, y} | x ∈ A, y ∈ B } A × B = { {x, y} | x ∈ A, y ∈ B }
``` ```
## "Generalized" operations ## "Generalized" operations
### General union ### General union
Better known as "flattening" of a set of sets. Better known as "flattening" of a set of sets.
``` ```
∪A = {x|X∈A,x∈X} ∪A = {x|X∈A,x∈X}
∪A={a,b,c,d,e,f} ∪A={a,b,c,d,e,f}
∪B={a} ∪B={a}
∪C=a∪{c,d} ∪C=a∪{c,d}
``` ```
### General intersection ### General intersection
``` ```
∩ A = A1 ∩ A2 ∩ … ∩ An ∩ A = A1 ∩ A2 ∩ … ∩ An
``` ```