N.B.

Iverson's Convention

As is customary for many mathematical notations in CS, this system uses Iverson's convention (Iverson's brackets) to denote certain conditional expressions. It should be understood that the notation will produce a value of \( 1 \) if the expression is true; otherwise, it will be strongly \( 0 \) --- that is, even if the expression would be undefined, it will still yield \( 0 \).

\( [ 1 \gt 0 ] = 1 \); \( [ 0 = 1 ] = 0 \); \( [ 5 \textrm{ is prime} ] = 1 \);

\( \sum \limits_{1 \leq k \leq 5} k = \sum \limits_k k [ 1 \leq k \leq 5 ] \)

Sets

In the equations represented above, it is to be assumed that undefined values in a set are implicitly 0; this simplifies the representations of the various summations; they are not intended to be vigorous.

For example: let \( x \) = \( \{ 1, 2, 3 \} \). Given the equation \( \sum_k x_k \), it is assumed that the solution is \( 1 + 2 + 3 = 6 \), not undefined. Formally, the former sum is to be interpreted as: \( \sum_{k=0}^n x_k \) where \( n \) is the length of set \( x \), or \( \sum_k x_k [x_k \textrm{ is defined}] \) using Iverson's convention (the latter of which our first notation is based upon by simply omitting the brackets and implying their existence).

Counting Sets

Let \(N(S)\) = the number of values within the set \(S\); this notation is used within certain summations. You may also see the following notations:

• \(\sum_{k} S_k\) to count the number of one-values in boolean set \(S\) (e.g. if \(S\) denotes properties with swimming pools, we can count the number of swimming pools).
• \(\sum_{k=0}^{N(S)} 1\) to count the number of values in set \(S\).

Vector Arithmetic

Only one type of vector arithmetic (dot products) is currently supported, but others may be done manually using sums and products. Dot products are denoted by \(a\cdot b\), where \(a\) and \(b\) are vectors.

Subscript Precedence

Subscripts should be applied from right to left. That is: \(S_{x_{y_z}}\) = \(S_{(x_{(y_z)})}\). In the event where a notation may be ambiguous (e.g. \(\theta_{1_x}\), since \(1_x\) could not possibly make sense in the context of this system), parenthesis will always be added to clarify intent.