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| diff --git a/set-theory.html.markdown b/set-theory.html.markdown index 988c4397..c6bc39c5 100644 --- a/set-theory.html.markdown +++ b/set-theory.html.markdown @@ -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∈A∨x∈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  ``` | 
