i'm just worried that the fact that my consequent is essentially included in my antecedent invalidates the proof, correct?
i have written an inductive proof of the statement that for any set
I am using my own notation for denoting set union with a disjoint set of cardinality 1, because it is such a convenience. is there some standard way of doing this? any tips/corrections, mathematical, stylistic, or notational will be highly appreciated.
denote where is some set disjoint from and where , so that we can write
The following holds trivially for and we will assume it to hold for any arbitrary set as well
thus the following also holds for any
we take as our inductive hypothesis that
recall that for any set we assume
which we can rearrange as follows
which proves the inductive step and the theorem.
thank again for any tips