Addition/Subtraction Principle
Addition Principle and Disjoint Unions
Definition: Two sets are said to be disjoint if \(X\cap Y=\emptyset\). Similarly, a collection \(X_1,\ldots, X_n\) of sets are said to be disjoint if any pair of them is disjoint.
Note that one can have 3 sets \(A\), \(B\), and \(C\) with \(A\cap B\cap C=\emptyset\) but \(A\), \(B\), and \(C\) are not disjoint.
Proposition: Let \(X_1,X_2,\ldots, X_n\) be a disjoint collection of finite sets. Then \[ |X_1\cup X_2\cup\cdots\cup X_n|=\sum_{i=1}^{n} |X_{i}|. \]
Proposition: Suppose that \(U\subset X\). Then \(|X-U|=|X|-|U|\)
Problem 3
(Problem 3 from section 3.3) Five cards are dealt from a 52-card deck and lined up in a row. How many such lineups are there in which all five cards are the same color (i.e. black or red)?
Problem 9
(Problem 9 from Section 3.3) Consider “words” of length 6 made from the letters \(A,B,C,D,E,F,G,H\). How many such words are possible if each letter can occur at most one time, and the word must contain two consecutive vowels?