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 ```