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,…}`
* `Q`, the set of all rational numbers.
* `R`, the set of all real numbers.
### The empty set
* The set containing no items is called the empty set. Representation: `∅`
* The empty set can be described as `∅ = {x|x ≠ x}`
* The empty set is always unique.
* The empty set is the subset of all sets.
```
A = {x|x∈N,x < 0}
A = ∅
@ -42,6 +44,7 @@ A = ∅
|∅| = 0
|{∅}| = 1
```
## Representing sets
### Enumeration
* List all items of the set, e.g. `A = {a,b,c,d}`
@ -49,6 +52,7 @@ A = ∅
### Description
* Describes the features of all items in the set. Syntax: `{body|condtion}`
```
A = {x|x is a vowel}
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|`
* 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.
```
A = {A,B,C}
|A| = 3
@ -94,6 +99,7 @@ B = {a,{b,c}}
### Powerset
* 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}
@ -103,41 +109,54 @@ P(A) = {x|x ⊆ A}
## Set operations among two sets
### 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`.
```
A B = {x|x∈Ax∈B}
```
### 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`.
```
A ∩ B = {x|x∈A,x∈B}
```
### 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`.
```
A \ B = {x|x∈A,x∉B}
```
### 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.
```
A △ B = {x|(x∈A∧x∉B)(x∈B∧x∉A)}
A △ B = (A \ B) (B \ A)
```
### 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`.
```
A × B = { {x, y} | x ∈ A, y ∈ B }
```
## "Generalized" operations
### General union
Better known as "flattening" of a set of sets.
```
A = {x|X∈A,x∈X}
A={a,b,c,d,e,f}
B={a}
C=a{c,d}
```
### General intersection
```
∩ A = A1 ∩ A2 ∩ … ∩ An
```