lancejpollard · March 15, 2012 18:12 · May 1, 2012 · Mar 17, 2012 · Mar 17, 2012 · Mar 17, 2012
diff --git a/index.md b/index.md
@@ -208,7 +208,9 @@ Cardinality.  The number of items in the set.
 
 ### `A×B`
 
-Cross product.
+Cross product.  The cross product is a binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which is perpendicular to the plane containing the two input vectors.
+
+http://stackoverflow.com/questions/243945/calculating-a-2d-vectors-cross-product
 
 ``` coffeescript
 # A = {1, 2}

diff --git a/index.md b/index.md
@@ -274,6 +274,18 @@ f(g(x)) #=> 26
 
 ### `∃! x: P(x)`
 
+### `P(A|B)`
+
+The probability of the event A occurring given that B occurs.
+
+### `A ⊥ B`
+
+A is an event whose probability is independent of event B.
+
+### `A ∝ B`
+
+The problem A can be polynomially reduced to the problem B.
+
 ### `W⊥`
 
 `W⊥` means the orthogonal complement of W (where W is a subspace of the inner product space V), the set of all vectors in V orthogonal to every vector in W.

diff --git a/index.md b/index.md
@@ -381,6 +381,24 @@ transposeMatrix([[1, 2, 3], [4, 5, 6]])
   #=> [[1, 4], [2, 5], [3, 6]]
 ```
 
+### Binomial Coefficient
+
+``` coffeescript
+binomial = (n, k) ->
+  coeff = 1
+  i     = n - k
+
+  coeff *= i while i++ < n
+
+  i     = 1
+
+  coeff /= i while i++ < k
+
+  coeff
+
+alert binomial(5, 2)
+```
+
 ## Library?
 
 It'd be cool if there was a library that could parse TeX-like [jqmath](http://mathscribe.com/author/jqmath.html) format into code:

diff --git a/index.md b/index.md
@@ -133,6 +133,10 @@ A = [1, 2, 3]
 B = [1, 4, 5, 6]
 ```
 
+### `ℝ2`
+
+The two-dimensional [vector space](http://en.wikipedia.org/wiki/Vector_space) of real numbers in mathematics.
+
 ### `ℕ`, `N`
 
 Natural numbers.  Integers that can physically exist, i.e. only positive integers.  Has different meanings.  Means either `{ 0, 1, 2, 3, …}` or `{ 1, 2, 3, …}`.

diff --git a/index.md b/index.md
@@ -270,6 +270,12 @@ f(g(x)) #=> 26
 
 ### `∃! x: P(x)`
 
+### `W⊥`
+
+`W⊥` means the orthogonal complement of W (where W is a subspace of the inner product space V), the set of all vectors in V orthogonal to every vector in W.
+
+<img src='http://upload.wikimedia.org/wikipedia/en/math/a/0/e/a0eb2e83f679f2dbb70e61a9182eb57a.png' />
+
 ### `||x||`
 
 Norm of the element x.  The length of the array.  Always positive.

diff --git a/index.md b/index.md
@@ -1,4 +1,6 @@
-## Sets == Arrays
+## Sets =~ Arrays
+
+A set is basically an array of unique items.
 
 ### `(A,B)`
 

diff --git a/index.md b/index.md
@@ -311,6 +311,64 @@ exampleSum = (n) ->
 exampleSum(10) #=> 605
 ```
 
+### Matrix Addition
+
+The sum `A+B` of two m-by-n matrices `A` and `B` is calculated with:
+
+`(A + B)i,j = Ai,j + Bi,j`, where `1 ≤ i ≤ m` and `1 ≤ j ≤ n`.
+
+``` coffeescript
+addMatrices = (a, b) ->
+  result = []
+
+  for rowA, rowI in a
+    rowB      = b[rowI]
+    rowResult = []
+
+    for columnA, columnI in rowA
+      columnB = rowB[columnI]
+      columnResult = columnA + columnB
+      rowResult.push(columnResult)
+
+    result.push(rowResult)
+
+  result
+
+addMatrices([[1, 2], [3, 4]], [[5, 6], [7, 8]])
+  #=> [[1 + 5, 2 + 6], [3 + 7, 4 + 8]]
+  #=> [[6, 8], [10, 12]]
+```
+
+### Matrix Transposition
+
+The transpose of an m-by-n matrix A is the n-by-m matrix `AT` formed by turning rows into columns and vice versa:
+
+`(AT)i,j = Aj,i`
+
+``` coffeescript
+transposeMatrix = (matrix) ->
+  result        = []
+  columnCount   = matrix[0].length
+  rowCount      = matrix.length
+
+  # construct new arrays
+  i             = 0
+  while i < columnCount
+    column      = []
+
+    result[i]   = column
+    i += 1
+
+  for row, i in matrix
+    for column, j in row
+      result[j][i] = column
+
+  result
+
+transposeMatrix([[1, 2, 3], [4, 5, 6]])
+  #=> [[1, 4], [2, 5], [3, 6]]
+```
+
 ## Library?
 
 It'd be cool if there was a library that could parse TeX-like [jqmath](http://mathscribe.com/author/jqmath.html) format into code:

diff --git a/index.md b/index.md
@@ -291,6 +291,10 @@ innerProduct = (u, v) ->
 innerProduct([2, 3], [-1, 5]) #=> 13
 ```
 
+### `det(A)`, `|A|`
+
+Determinant of a matrix.
+
 ## Examples
 
 ### `$$∑↙{i=0}↖n i={n(n+1)}/2$$`

diff --git a/index.md b/index.md
@@ -212,6 +212,7 @@ Cross product.
 # [[1, x], [1, y], [1, z], [2, x], [2, y], [2, z]]
 A = [1, 2]
 B = [x, y, z]
+# crossProduct function???      
 ```
 
 ### `A·B`

diff --git a/index.md b/index.md
@@ -267,6 +267,29 @@ f(g(x)) #=> 26
 
 ### `∃! x: P(x)`
 
+### `||x||`
+
+Norm of the element x.  The length of the array.  Always positive.
+
+``` coffeescript
+[1, 2, 3].length #=> 3
+```
+
+### `⟨u,v⟩`
+
+The inner product of u and v, where u and v are members of an inner product space.
+
+``` coffeescript
+# x = (2, 3)
+# y = (−1, 5)
+# ⟨x, y⟩ = 2 × −1 + 3 × 5 = 13
+
+innerProduct = (u, v) ->
+  crossProduct(u) + crossProduct(v)
+
+innerProduct([2, 3], [-1, 5]) #=> 13
+```
+
 ## Examples
 
 ### `$$∑↙{i=0}↖n i={n(n+1)}/2$$`

diff --git a/index.md b/index.md
@@ -216,15 +216,15 @@ B = [x, y, z]
 
 ### `A·B`
 
-Cross product.
+Dot product.
 
 <img src='http://upload.wikimedia.org/wikipedia/en/math/2/1/4/21467e485cbafd1a308e388211257ad6.png' />
 
 ``` coffeescript
 # one dimensional vector
 A = [1, 2, 3]
 B = [4, 5, 6]
-crossProduct = (a, b) =>
+dotProduct = (a, b) =>
   n      = a.length # or b.length
   i      = 0
   result = 0

diff --git a/index.md b/index.md
@@ -214,6 +214,28 @@ A = [1, 2]
 B = [x, y, z]
 ```
 
+### `A·B`
+
+Cross product.
+
+<img src='http://upload.wikimedia.org/wikipedia/en/math/2/1/4/21467e485cbafd1a308e388211257ad6.png' />
+
+``` coffeescript
+# one dimensional vector
+A = [1, 2, 3]
+B = [4, 5, 6]
+crossProduct = (a, b) =>
+  n      = a.length # or b.length
+  i      = 0
+  result = 0
+
+  while i < n
+    result += (a[i] * b[i])
+    i += 1
+
+  result
+```
+
 ### `f: X → Y`
 
 Function f maps the set X into the set Y. Or, there is a function f such that X is converted to Y.

diff --git a/index.md b/index.md
@@ -16,6 +16,17 @@ Unordered set.
 {"a": 1, "b": 2} == {"b": 2, "a": 1}
 ```
 
+### `A ∆ B`
+
+Symmetric difference.  The set of elements in exactly one of A or B.  The unique items when you combine A and B.
+
+``` coffeescript
+# {1,5,6,8} ∆ {2,5,8} = {1,2,6}
+A = [1, 5, 6, 8]
+B = [2, 5, 8]
+# A ∆ B = [1, 2, 6]
+```
+
 ### `A ⊇ B`
 
 A is a superset of B.  The A array has all items the B array has.

diff --git a/index.md b/index.md
@@ -228,6 +228,12 @@ x = 10
 f(g(x)) #=> 26
 ```
 
+### `∀ x: P(x)`
+
+### `∃ x: P(x)`
+
+### `∃! x: P(x)`
+
 ## Examples
 
 ### `$$∑↙{i=0}↖n i={n(n+1)}/2$$`

diff --git a/index.md b/index.md
@@ -5,7 +5,7 @@
 Ordered set.
 
 ``` coffeescript
-[1, 2] != [1, 2]
+[1, 2] != [2, 1]
 ```
 
 ### `{A,B}`

diff --git a/index.md b/index.md
@@ -194,6 +194,8 @@ Cardinality.  The number of items in the set.
 Cross product.
 
 ``` coffeescript
+# A = {1, 2}
+# B = {x, y, z}
 # A×B = {(a, b) : a  A, b  B}
 # A×B = {(1, x), (1, y), (1, z), (2, x), (2, y), (2, z)}
 # [[1, x], [1, y], [1, z], [2, x], [2, y], [2, z]]

diff --git a/index.md b/index.md
@@ -203,7 +203,7 @@ B = [x, y, z]
 
 ### `f: X → Y`
 
-Function f maps the set X into the set Y.
+Function f maps the set X into the set Y. Or, there is a function f such that X is converted to Y.
 
 ``` coffeescript
 X = [1, 2, 3]

diff --git a/index.md b/index.md
@@ -201,7 +201,7 @@ A = [1, 2]
 B = [x, y, z]
 ```
 
-## `f: X → Y`
+### `f: X → Y`
 
 Function f maps the set X into the set Y.
 
@@ -212,6 +212,20 @@ f = array.reverse
 f(X) == Y
 ```
 
+### `(f∘g)(x)`
+
+Function composition.
+
+``` coffeescript
+# f(x) := 2x
+# g(x) := x + 3
+# (f∘g)(x) = 2(x + 3)
+f = (x) -> 2 * x
+g = (x) -> x + 3
+x = 10
+f(g(x)) #=> 26
+```
+
 ## Examples
 
 ### `$$∑↙{i=0}↖n i={n(n+1)}/2$$`

diff --git a/index.md b/index.md
@@ -201,6 +201,17 @@ A = [1, 2]
 B = [x, y, z]
 ```
 
+## `f: X → Y`
+
+Function f maps the set X into the set Y.
+
+``` coffeescript
+X = [1, 2, 3]
+Y = [3, 2, 1]
+f = array.reverse
+f(X) == Y
+```
+
 ## Examples
 
 ### `$$∑↙{i=0}↖n i={n(n+1)}/2$$`

diff --git a/index.md b/index.md
@@ -189,6 +189,18 @@ Cardinality.  The number of items in the set.
 [3, 5, 7, 9].length
 ```
 
+### `A×B`
+
+Cross product.
+
+``` coffeescript
+# A×B = {(a, b) : a  A, b  B}
+# A×B = {(1, x), (1, y), (1, z), (2, x), (2, y), (2, z)}
+# [[1, x], [1, y], [1, z], [2, x], [2, y], [2, z]]
+A = [1, 2]
+B = [x, y, z]
+```
+
 ## Examples
 
 ### `$$∑↙{i=0}↖n i={n(n+1)}/2$$`

diff --git a/index.md b/index.md
@@ -1,5 +1,21 @@
 ## Sets == Arrays
 
+### `(A,B)`
+
+Ordered set.
+
+``` coffeescript
+[1, 2] != [1, 2]
+```
+
+### `{A,B}`
+
+Unordered set.
+
+``` coffeescript
+{"a": 1, "b": 2} == {"b": 2, "a": 1}
+```
+
 ### `A ⊇ B`
 
 A is a superset of B.  The A array has all items the B array has.
@@ -276,4 +292,5 @@ P = subsetGreaterThanZero([-1, 0, 1, 2]) #=> [1, 2]
 - http://en.wikipedia.org/wiki/List_of_mathematical_symbols
 - http://www.esi2.us.es/~mbilbao/pdffiles/DiestelGT.pdf
 - http://mathscribe.com/author/jqmath.html
-- http://en.wikipedia.org/wiki/Help:Displaying_a_formula
+- http://en.wikipedia.org/wiki/Help:Displaying_a_formula
+- http://plato.stanford.edu/entries/set-theory/primer.html
diff --git a/index.md b/index.md
@@ -258,6 +258,19 @@ graph =
 averageDegree(graph) #=> 4
 ```
 
+### `P = {G | X(G) > 0}`
+
+``` coffeescript
+x = (set) ->
+  # magic!
+  set.length
+
+subsetGreaterThanZero = (G) ->
+  x(G) > 0
+
+P = subsetGreaterThanZero([-1, 0, 1, 2]) #=> [1, 2]
+```
+
 ## Resources
 
 - http://en.wikipedia.org/wiki/List_of_mathematical_symbols

diff --git a/index.md b/index.md
@@ -262,4 +262,5 @@ averageDegree(graph) #=> 4
 
 - http://en.wikipedia.org/wiki/List_of_mathematical_symbols
 - http://www.esi2.us.es/~mbilbao/pdffiles/DiestelGT.pdf
-- http://mathscribe.com/author/jqmath.html
+- http://mathscribe.com/author/jqmath.html
+- http://en.wikipedia.org/wiki/Help:Displaying_a_formula
diff --git a/index.md b/index.md
@@ -229,6 +229,10 @@ So, for each item in the array, execute the function: `d(v)`.  Simple as that, t
 
 All together.  The degree of a graph is [arbitrarily] defined by this: for each vertex in the graph, count the number of edges, and divide by the number of vertices.  That is, it's the average number of edges per vertex!  Are you kidding me!  All that just for a simple average?
 
+Here's the equation again:
+
+<img src='http://i.imgur.com/Wq3BO.png'/>
+
 In code, that looks like this:
 
 ``` coffeescript

diff --git a/index.md b/index.md
@@ -219,7 +219,7 @@ alert math("[1, 2, 3] ∪ [1, 2, 3, 4, 5]")
 
 What is this saying (from the [Graph Theory](http://www.esi2.us.es/~mbilbao/pdffiles/DiestelGT.pdf) book)??
 
-Well, `:=` means "I want to call the thing on the right the thing on the left".  So call that `∑` equation `d(g)`.  It's arbitrary.  He's saying `d(v)` is going to be called the "degree" of a vertex, which he's defining as "the number of edges on that vertex".  Then he's saying `d(G)` is going to be called the "degree" of the whole graph.
+Well, `:=` means "I want to call the thing on the right the thing on the left".  So call that `∑` equation `d(g)`.  It's arbitrary.  He's saying `d(v)` is going to be called the "degree" of a vertex, which he's defining as "the number of edges on that vertex".  Then he's saying `d(G)` is going to be called the "degree" of the whole graph.  He's basically just naming his functions and variables!
 
 Then `1/|V|`.  He defines `V` to be the number of vertices in a graph.  So, number of items in an array, or `array.length`, so `1/vertices.length`.
 

diff --git a/index.md b/index.md
@@ -219,7 +219,7 @@ alert math("[1, 2, 3] ∪ [1, 2, 3, 4, 5]")
 
 What is this saying (from the [Graph Theory](http://www.esi2.us.es/~mbilbao/pdffiles/DiestelGT.pdf) book)??
 
-Well, `:=` means "I want to call the thing on the right the thing on the left".  So call that `∑` equation `d(g)`.  It's arbitrary.  He's saying `d(v)` is going to be called the "degree" of a vertex, which he means "the number of edges on that vertex".  Then he's saying `d(G)` is going to be called the "degree" of the whole graph.
+Well, `:=` means "I want to call the thing on the right the thing on the left".  So call that `∑` equation `d(g)`.  It's arbitrary.  He's saying `d(v)` is going to be called the "degree" of a vertex, which he's defining as "the number of edges on that vertex".  Then he's saying `d(G)` is going to be called the "degree" of the whole graph.
 
 Then `1/|V|`.  He defines `V` to be the number of vertices in a graph.  So, number of items in an array, or `array.length`, so `1/vertices.length`.
 

diff --git a/index.md b/index.md
@@ -213,7 +213,9 @@ math = (string) ->
 alert math("[1, 2, 3] ∪ [1, 2, 3, 4, 5]")
 ```
 
-<h3>Average Degree of a Graph: <a name='graph-theory-equation' href='#graph-theory-equation'><img src='http://i.imgur.com/Wq3BO.png'/></a></h3>
+<h3><a name='graph-theory-equation' href='#graph-theory-equation'>#</a>Average Degree of a Graph</h3>
+
+<img src='http://i.imgur.com/Wq3BO.png'/>
 
 What is this saying (from the [Graph Theory](http://www.esi2.us.es/~mbilbao/pdffiles/DiestelGT.pdf) book)??
 

diff --git a/index.md b/index.md
@@ -213,7 +213,7 @@ math = (string) ->
 alert math("[1, 2, 3] ∪ [1, 2, 3, 4, 5]")
 ```
 
-<h3><a name='graph-theory-equation' href='#graph-theory-equation'><img src='http://i.imgur.com/Wq3BO.png'/></a></h3>
+<h3>Average Degree of a Graph: <a name='graph-theory-equation' href='#graph-theory-equation'><img src='http://i.imgur.com/Wq3BO.png'/></a></h3>
 
 What is this saying (from the [Graph Theory](http://www.esi2.us.es/~mbilbao/pdffiles/DiestelGT.pdf) book)??
 

diff --git a/index.md b/index.md
@@ -213,7 +213,7 @@ math = (string) ->
 alert math("[1, 2, 3] ∪ [1, 2, 3, 4, 5]")
 ```
 
-<h3><a name='graph-theory-equation' href='name='#graph-theory-equation'><img src='http://i.imgur.com/Wq3BO.png'/></a></h3>
+<h3><a name='graph-theory-equation' href='#graph-theory-equation'><img src='http://i.imgur.com/Wq3BO.png'/></a></h3>
 
 What is this saying (from the [Graph Theory](http://www.esi2.us.es/~mbilbao/pdffiles/DiestelGT.pdf) book)??
 

diff --git a/index.md b/index.md
@@ -213,7 +213,7 @@ math = (string) ->
 alert math("[1, 2, 3] ∪ [1, 2, 3, 4, 5]")
 ```
 
-<h3 id='graph-theory-equation'><img src='http://i.imgur.com/Wq3BO.png'/></h3>
+<h3><a name='graph-theory-equation' href='name='#graph-theory-equation'><img src='http://i.imgur.com/Wq3BO.png'/></a></h3>
 
 What is this saying (from the [Graph Theory](http://www.esi2.us.es/~mbilbao/pdffiles/DiestelGT.pdf) book)??