Sets – Quick Revision Bullets

Fundamental Concepts

• Set: a well-defined collection of objects; elements denoted by lower-case letters, sets by capital letters.
• Membership: $a \in A$ (belongs), $b \notin A$ (does not belong).
• Representations
  • Roster (tabular): list elements inside braces ${2,4,6}$
  • Set-builder: describe common property ${x : x\text{ is even}, x<7}$
• Repetition & order of elements are irrelevant in a set.

Standard Number Sets

• $\mathbb N$ natural, $\mathbb Z$ integers, $\mathbb Q$ rationals, $\mathbb R$ reals.
• Positive subsets: $\mathbb Z^+, \mathbb Q^+, \mathbb R^+$.

Special Kinds of Sets

• Empty (null) set: $\varnothing = {}$, contains no elements.
• Singleton: exactly one element ${a}$.
• Finite set: $n(A) \in \mathbb N\cup{0}$; otherwise infinite.
• Equal sets: $A=B$ iff every element of one is in the other.
• Subset: $A\subseteq B\iff x\in A\Rightarrow x\in B$; proper subset $A\subset B$ when $A\neq B$.
• Universal set $U$: context-dependent superset of all sets under discussion.

Intervals inside $\mathbb R$

• Open: $(a,b)={x:a<x<b}$
• Closed: $[a,b]={x:a\le x\le b}$
• Half-open: $[a,b)$ and $(a,b]$
• Length of any above interval: $b-a$.

Venn Diagram Basics

• Rectangle represents $U$; circles represent subsets.
• Useful for visualising union, intersection, difference, complement.

Operations on Sets

• 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 (relative to $U$): $A' = U-A = {x:x\notin A}$

Key Laws & Properties

• Union

• Commutative $A\cup B=B\cup A$
• Associative $(A\cup B)\cup C=A\cup(B\cup C)$
• Identity $A\cup\varnothing=A$
• Idempotent $A\cup A=A$
  • Intersection
• Commutative $A\cap B=B\cap A$
• Associative $(A\cap B)\cap C=A\cap(B\cap C)$
• Identity $A\cap U=A$
• Idempotent $A\cap A=A$
• Distributive $A\cap(B\cup C)=(A\cap B)\cup(A\cap C)$ (and union distributes over intersection analogously)
  • Complement
• $A\cup A' = U$, $A\cap A' = \varnothing$
• Double complement $(A')' = A$
• De Morgan: $(A\cup B)' = A'\cap B'$, $(A\cap B)' = A'\cup B'$
  • Subset criteria equivalences
• $A\subseteq B \iff A-B=\varnothing \iff A\cup B=B \iff A\cap B=A$
  • Disjoint sets: $A\cap B=\varnothing$.

Quick Facts

• $\varnothing$ is a subset of every set.
• Every set is a subset of itself.
• If $B\subseteq A$ then $A\cup B = A$ and $A\cap B = B$.
• Standard chain inside $\mathbb R$: $\mathbb N \subset \mathbb Z \subset \mathbb Q \subset \mathbb R$; irrationals $\mathbb R\setminus\mathbb Q$.