Friday, January 24, 2014

COVER math

I had this idea for something I call COVER math, which is an acronym that stands for constants, operands, variables, expressions, replacements.
For constants you have a set perhaps infinite of values that I will write as lower case letters {a,b,c,...}
For variables you have a set perhaps infinite of symbols that can represent any value in the above set, written with Greek letters {alpha, beta, gamma, ...}
For operands you have a set perhaps infinite of one to one and onto relations from the set of constants to the set of constants represented by upper case letters {A,B,C,...}
An expression is any combination of operands working on variables or constants with parenthesis suggesting the order of operation e.g.
would be the constant a and b under the operand A, the result of which is under operand B with variable alpha. 
A replacement is a list of rules for rewriting an expression for example:

Ordinary algebra is a COVER math, using the real numbers as constants, variables having their usual meaning, operands being the usual plus, minus, multiplication, and division, and operations such as distributivity of multiplication written as replacements. 

A question is every type of math and logic a COVER math? 

2. A simple but much different example could be constants {a,b,c,d,e}, operands {A,B,C}, variable {Alpha}, and no replacement rules. to define the operands we can use a set of tables from {a,b,c,d,e}X{a,b,c,d,e} such as the following...
An example expression:
simplifies to:
and can be graphed as:
with a domain over alpha and range in {a,b,c,d,e}

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