Summary

This chapter built the language of sets from the ground up.

• A set is a well-defined collection of distinct objects, written in roster form (listing elements) or set-builder form (stating a defining property).
• Sets are classified as the empty set $\varnothing$ (no elements), finite or infinite sets (by whether their elements can be exhaustively counted), and two sets are equal when they contain exactly the same elements.
• $A$ is a subset of $B$ ($A \subseteq B$) when every element of $A$ lies in $B$; a proper subset ($A \subset B$) additionally requires $B$ to have some element outside $A$. Subsets of $\mathbb{R}$ of the form "between two numbers" are written compactly as intervals: $(a,b)$, $[a,b]$, $[a,b)$, $(a,b]$, and their unbounded versions with $\pm\infty$.
• The power set $P(A)$ collects all subsets of $A$, with $n(P(A)) = 2^{n(A)}$. The universal set $U$ is the fixed "master set" that frames every other set in a problem, and is essential for defining the complement.
• Venn diagrams picture $U$ as a rectangle and every subset as a circle inside it, giving an immediate visual reading of union, intersection, difference and complement.
• $A \cup B$ collects everything in either set; $A \cap B$ keeps only what is common to both; disjoint sets have $A \cap B = \varnothing$; and $n(A \cup B) = n(A) + n(B) - n(A \cap B)$ for finite sets.
• $A - B$ keeps only $A$'s elements that are absent from $B$; it is not commutative ($A - B \ne B - A$ in general), and $A - B$, $B - A$ are always disjoint.
• The complement $A' = U - A$ satisfies $A \cup A' = U$, $A \cap A' = \varnothing$, $(A')' = A$, and De Morgan's laws $(A \cup B)' = A' \cap B'$, $(A \cap B)' = A' \cup B'$ — both proved directly from …