The event consisting of all outcomes that are in either A or B.
Notation: \(A \cup B\) (Reads: A union B)
Roll a six-sided die 🎲
\[ \begin{aligned} A &= \{ 5, 6 \} \\ \\ B &= \{ 1, 3, 5 \} \\ \\ A \cup B &= \\ \end{aligned} \]
The event consisting of all outcomes that are in both A and B.
Notation: \(A \cap B\) (Reads: A intersect B)
Roll a six-sided die 🎲
\[ \begin{aligned} A &= \{ 5, 6 \} \\ \\ B &= \{ 1, 3, 5 \} \\ \\ A \cap B &= \end{aligned} \]
\[ a + b = b + a \]
\[ a \times b = b \times a \]
\[ A \cup B = B \cup A \]
\[ A \cap B = B \cap A \]
\[ (a + b) + c = a + (b + c) \]
\[ (a \times b) \times c = a \times (b \times c) \]
\[ (A \cup B) \cup C = A \cup (B \cup C) \]
\[ (A \cap B) \cap C = A \cap (B \cap C) \]
\[ (a + b) \times c = (a \times c) + (b \times c) \]
\[ (A \cup B) \cap C = (A \cap C) \cup (B \cap C) \]
\[ (A \cap B) \cup C = (A \cup C) \cap (B \cup C) \]
Null event \(\Phi\): the event consisting of no outcomes.
\(A\) and \(B\) are mutually exclusive (or disjoint) events if they do not share any outcome, or simply
\[A \cap B=\Phi\]
Roll a six-sided die 🎲
\[ \begin{aligned} A &= \{ 1, 3, 5\} \text{ (odd numbers)} \\ B &= \{ 2, 4, 6 \} \text{ (even numbers)} \\ A \cap B &= \Phi \\ \end{aligned} \]
The set of all outcomes in \(\Omega\) that are not in A.
Notation: \(A^c\) (Reads: A complement)
By definition,
\[A \cap A^c = \Phi\]
\[A \cup A^c = \Omega\]
Roll a six-sided die 🎲
\[ \begin{aligned} A &= \{ 5, 6 \} \\ A^c &= \\ \end{aligned} \]
An event A is a subset of an event B if all outcomes in A are also outcomes in B.
Notation: \(A \subseteq B\)
Roll a six-sided die 🎲
\[ \begin{aligned} A &= \{ 1, 3, 5\} \text{ (odd numbers)} \\ \\ B &= \{ 1, 2, 3, 4, 5 \} \text{ (less than 6)} \\ \\ A &\subseteq B \\ \end{aligned} \]
What is the red region in each Venn diagram?
\[ (A \cup B)^c = A^c \cap B^c \]
\[ (A \cap B)^c = A^c \cup B^c \]
\[ A \cap B = \Phi \]
\[ A \subseteq B \]