Addition Rule

The addition rule says that if you have a set of \(n\) items, and a set of \(m\) different items, then if you combine the sets you will have \(n + m\) items.

Example: Suppose you go to one store that sells brown and black basketballs. You have 2 color options in the first store. Then, in a second store, you find colorful basketballs. There is a red, a purple, and a green ball. You have 3 color options in the second store.

You have 2 options in the first store and 3 options in the second store. So, by the addition rule, you have a total of 5 options:

1. Brown
2. Black
3. Red
4. Purple
5. Green

Addition rule stated for sets: If \(A\) and \(B\) are disjoint, finite sets, then \(|A \cup B| = |A| + |B|.\)

Example: Let \(A = \{1, 3, 5\}\) and \(B = \{2, 8, 10, 11\}.\) Then \(|A| = 3\) and \(|B| = 4.\)

The union is \(A \cup B = \{1,2,3,5,8,10,11\},\) so \(|A \cup B| = 7.\) Since \(3 + 4 = 7,\) \(|A|+|B| = |A \cup B|.\)