An index set is a set used for indexing, or labeling.
Numbers, sets, or even objects can be indexed. Suppose we want to list the colors red, green, and blue as color 1, color 2, and color 3. The labels 1, 2, and 3 are the index set: \[I = \{1,2,3\}\]
Example
Example: Your friend is playing 4 games numbered game 1 through game 4. For each game, there is a chance to win a prize. The prize for game 1 is $4, the prize for game 2 is $6, the prize for game 3 is $12, and the prize for game 4 is $15.
To compute the amount of money your friend wins, let \(I = \{1, 2, 3, 4\}\) be the set that corresponds to games 1, 2, 3 and 4. We will use \(I\) as the index set for the prize money. Let \(p_1, p_2, p_3,\) and \(p_4\) be the prize money for games 1, 2, 3, and 4, respectively. So, \(p_1 = 4,\) \(p_2 = 6,\) \(p_3 = 12,\) and \(p_4 = 15.\)
Let \(W\) be the set of games that your friend wins. The amount of prize money that your friend wins is the sum over the set \(\{p_i : i \in W\}.\) If your friend only wins the prize for games 1, 2 and 4, then \(W = \{1, 2, 4\}.\) So, the amount your friend wins is \[p_1 + p_2 + p_4 = 4 + 6 + 15 = 25\]
Your friend will get a $25 prize for winning games 1, 2, and 4.
Infinity
When the index set is \(I = \{1, 2, 3, \dots\},\) the index set is labeled from 1 to infinity. For example, to write the sum of the numbers \(a_1, a_2, a_3, \dots,\) we write
\[\sum_{i=1}^\infty a_i = a_1 + a_2 + a_3 + a_4 + \dots\]
To write the union of the sets \(A_1, A_2, A_3, \dots,\) we write
\[\bigcup_{i=1}^\infty A_i = A_1 \cup A_2 \cup A_3 \cup A_4 \cup \dots\]
Plugging elements in as a formula
When the elements can be written as a function of the index set, they are sometimes written as a formula.
For example, suppose the index set is \(I = \{2, 3, 4, 5\},\) and the numbers being indexed are \(a_2 = 2, a_3 = 3, a_4 = 4,\) and \(a_5 = 5.\) A concise way to write the set is \(a_i = i\) for every \(i \in I.\) Since \(a_i = i,\) the sum of the \(a_i\) can be written
\[\sum_{i=2}^5 i = 2 + 3 + 4 + 5 = 14\]
So, the value \(i\) can be plugged in directly for \(a_i.\)
To write the union over \(A_i = \{i, i+2\}\) for \(I = \{2, 3, 4, 5\},\) we write
\[\bigcup_{i=2}^5 \{i, i+2\} = \{2, 4\} \cup \{3, 5\} \cup \{4, 6\} \cup \{5, 7\} = \{3,4,5,6,7\}\]