📂Set Theory

Elementary Notation in Enumerating Elements

Explanation

$$\left\{ 1, \cdots , \hat{k} , \cdots , n \right\}$$

When representing a set, there are instances where a hat is placed on a specific index as shown above, which means that the element is to be removed. For instance, in $\left\{ 0,1,2,3,4 \right\}$, if we consider the collection of sets with exactly one element removed, it can be represented as $$\left\{ \left\{ 0, \cdots, \hat{k}, \cdots, 4 \right\} : j = 0,1,2,3,4 \right\}$$ This example is very simplistic for a finite set, but the hat notation is actually very useful in places where it is used as it can omit cumbersome expressions such as the difference set $\setminus \left\{ k \right\}$. It is not only used in sets, and can suddenly appear in places where an index appears like in the following. $$\sum_{i=1 , \cdots , \hat{k} , \cdots, n} {{ 1 } \over { i^{2} }}$$