2.2 Set theory and operations

Set theory forms the foundation for understanding probability. It provides tools to organize and analyze events in a sample space, crucial for calculating probabilities.

Operations like union, intersection, and complement help us combine or separate events. This allows us to solve complex probability problems by breaking them down into simpler parts.

Set Theory Fundamentals

Fundamentals of set theory

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• Set: collection of distinct objects or elements denoted by curly braces $\{1, 2, 3\}$
  • Order of elements does not matter (commutative property)
  • Duplicate elements are not allowed (idempotent property)
• Element: object or member of a set denoted by the symbol $\in$ ($1 \in \{1, 2, 3\}$)
• Universal set: set of all elements under consideration, denoted by $U$ or $\Omega$
  • Example: all natural numbers $\mathbb{N} = \{1, 2, 3, ...\}$
• Subset: set whose elements are all contained within another set, denoted by $\subseteq$ ($\{1, 2\} \subseteq \{1, 2, 3\}$)
  • Every set is a subset of itself (reflexive property)
  • Empty set $\emptyset$ or $\{\}$ is a subset of every set
• Proper subset: subset that is not equal to the original set, denoted by $\subset$ ($\{1, 2\} \subset \{1, 2, 3\}$)
• Equality of sets: two sets are equal if they have the same elements regardless of order ($\{1, 2, 3\} = \{3, 2, 1\}$)

Operations on sets

• Union: set containing all elements from two or more sets, denoted by $\cup$ ($\{1, 2\} \cup \{2, 3\} = \{1, 2, 3\}$)
  • Formula: $A \cup B = \{x | x \in A \text{ or } x \in B\}$
• Intersection: set containing elements common to two or more sets, denoted by $\cap$ ($\{1, 2\} \cap \{2, 3\} = \{2\}$)
  • Formula: $A \cap B = \{x | x \in A \text{ and } x \in B\}$
• Complement: set containing all elements in the universal set that are not in a given set, denoted by $A^c$ or $A'$
  • Formula: $A^c = \{x | x \in U \text{ and } x \notin A\}$
  • Example: if $U = \{1, 2, 3, 4\}$ and $A = \{1, 2\}$, then $A^c = \{3, 4\}$
• Difference: set containing elements in one set that are not in another set, denoted by $\setminus$ ($\{1, 2, 3\} \setminus \{2, 3\} = \{1\}$)
  • Formula: $A \setminus B = \{x | x \in A \text{ and } x \notin B\}$
• Symmetric difference: set containing elements that are in either of two sets but not in both, denoted by $\triangle$
  • Formula: $A \triangle B = (A \setminus B) \cup (B \setminus A)$
  • Example: $\{1, 2\} \triangle \{2, 3\} = \{1, 3\}$

Venn diagrams for sets

• Venn diagram: visual representation of set relationships using overlapping circles or shapes
  • Each set represented by a circle or shape
  • Overlapping regions represent shared elements
  • Non-overlapping regions represent unique elements
• Shading: indicates the region of interest in a Venn diagram
  • Example: shading the overlapping region of two sets represents their intersection
• Venn diagrams can illustrate:
  • Set operations (union, intersection, complement)
  • Subset and proper subset relationships
  • Mutually exclusive sets with no elements in common
  • Exhaustive sets whose union equals the universal set

Set theory in probability

• Event: subset of the sample space (universal set in probability)
• Probability of event A: measure of likelihood that event A will occur, denoted by $P(A)$
• Union of events: probability that at least one event occurs
  • Formula: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
• Intersection of events: probability that all events occur simultaneously
  • Formula: $P(A \cap B) = P(A) \times P(B|A)$, where $P(B|A)$ is conditional probability of B given A
• Complement of an event: probability that an event does not occur
  • Formula: $P(A^c) = 1 - P(A)$
• Mutually exclusive events: events that cannot occur simultaneously, $P(A \cap B) = 0$
  • For mutually exclusive events, $P(A \cup B) = P(A) + P(B)$
  • Example: rolling a 1 and rolling a 6 on a six-sided die
• Independent events: events that do not influence each other, $P(A \cap B) = P(A) \times P(B)$
  • Example: flipping a coin and rolling a die