What is the Correct Logical Statement for a Fractional Factorial Design?

[Alephu5]

I am writing a theorem to do with a fractional factorial design for an experiment. I have had minimal formal training in mathematics, and this is my first theorem. I am fairly happy with most of the statement, but the last part does not feel right.

Basically I want to say "If S is a subset of R, where the cardinal number of S is less than n(l-1)+1, then out of all possible sets composed of symmetric differences and unions (of the symmetric differences) between elements of S there does not exist a superset of P.

S⊂R:|S|<n(l-1)+1 ⇒∀{α│s_x ∆s_y ∧s_x∪s_y:(s_x∧s_y)∈S∨(s_x∧s_y )=s_x ∆s_y }∄α⊃P

Note: This is not the full theorem, I have defined n, l, R and P in a previous statement.

Can anyone confirm if this is correct, and if it isn't how I can correct it?

 

[Stephen Tashi]

Basically I want to say "If S is a subset of R, where the cardinal number of S is less than n(l-1)+1, then out of all possible sets composed of symmetric differences and unions (of the symmetric differences) between elements of S there does not exist a superset of P.

Unless you are writing a paper that deals with symbolic logic or mechanical theorem proving, etc. it isn't necessary to write mathematical statements in purely symbolic form. In fact, it is unwise to use symbols exclusively.

S⊂R:|S|<n(l-1)+1 ⇒∀{α│s_x ∆s_y ∧s_x∪s_y:(s_x∧s_y)∈S∨(s_x∧s_y )=s_x ∆s_y }∄α⊃P

You'll have to explain whether you have made any definitions that make your notation powerful enough to capture the concept of " all possible sets composed of symmetric differences and unions (of the symmetric differences) between elements of S". Your notation seems to say something about a set being "a symmetric difference or a symmetric union". I don't see where the thought of "all possible combinations of ..." is expressed.

Such a collection of sets is usually expressed by using a recursive definition.