Just a quick question here. I am working on an assignment for algorithms involving dynamic programming. Don't worry, this isn't a question about my assignment, just a question about understanding a portion of the original question. I am having difficulty understanding what this means:
$T(i, j) = 0 \,\forall\,i, \,j, \,j<i$
So far I understand: $T(i,j)$ is equal to 0 for all $i$. However, I am not certain how the $j$ and the $j < i$ come into play. Perhaps I just need some properly placed parentheses?
Any help would be great!
Formally (and IMHO always, if you want to avoid confusion), quantifiers should be placed at the beginning of the formula they govern, and should have only one quantified varibale each. Thus: $$\forall i\ \forall j\ \bigl(\ (\,j<i\,)\to (\,T(i,j)=0\,)\ \bigr)\ .$$ Better still, use words instead of symbols: $$\hbox{for all $i$ and for all $j$, if $j<i$ then $T(i,j)=0$.}$$