Take a sequence A. Imagine a sequence C=(An)nāJ, where J ā U. U is an index set. J ā U means J is a proper subsequence of U. Therefore, various possible sequences C are denoted by restricting the original sequence A to a smaller index set.
When C is defined as above, will simply showing the variable C inherently signify all possible sequences or just one possible sequence? How can I make sure to formally denote all possible sequences C can be? Is there a symbol commonly used for this?
Best Answer
I'm assuming you have defined $\mathbb A = (A_n)_{n\in \mathbf U}$. Then any subsequence of $\mathbb A$ can be defined as $$\mathbb C=(A_n)_{n\in \mathbf J},\mathbf J\subseteq \mathbf U$$ The collection of all possible subsequences of $\mathbb A$ is then given as $$\mathcal C=\{(A_n)_{n\in \mathbf J}: \mathbf J\subseteq \mathbf U \}$$