Mathematics · Ch 1 — Sets

Summary

A set is a well-defined collection of distinct objects, denoted by capital letters like $A$ , $B$ , etc.
Representation: Roster form (listing elements) and set-builder form (defining property).
Types of sets:
- Empty set: $\emptyset$ or $\{\}$
- Finite set: countable number of elements
- Infinite set: uncountable elements
- Equal sets: $A = B$ if they have exactly the same elements
- Subset: $A \subseteq B$ means every element of $A$ is in $B$
- Power set: $P(A)$ is the set of all subsets of $A$ ; if $|A| = n$ , then $|P(A)| = 2^n$
Set operations:
- Union: $A \cup B = \{x : x \in A \text{ or } x \in B\}$
- Intersection: $A \cap B = \{x : x \in A \text{ and } x \in B\}$
- Difference: $A - B = \{x : x \in A \text{ and } x \notin B\}$
- Complement: $A' = U - A$ (relative to universal set $U$ )
Key properties (for any sets $A, B, C$ $A, B, C$ ):
- Commutative: $A \cup B = B \cup A$ , $A \cap B = B \cap A$
- Associative: $(A \cup B) \cup C = A \cup (B \cup C)$ , similarly for $\cap$ …