Sets

Sets recap. A set is a well-defined, distinguishable collection of objects: given any object, we must be able to decide definitively whether it belongs to the collection or not. "The collection of all beautiful flowers in Ooty Rose Garden" is not a set (beauty is not sharply defined), but "the collection of all red flowers in Ooty Rose Garden" is a set. Likewise "old men in Tamil Nadu" is not well-defined, but "men in Tamil Nadu older than 70" is.

Membership across sets. The symbol $\in$ normally sits between an element and a set, but it is meaningful to write $A\in B$ when $A$ itself is an element of $B$ (i.e. $B$ contains the set $A$ as one of its members). For example, if $A=\{1,2\}$ and $B=\{1,\{1,2\},3,4\}$, then $A\in B$, because the object $\{1,2\}$ is literally one of $B$'s four listed elements.

Subsets.

• The empty set $\varnothing$ (or $\{\ \}$) has no elements.
• $A\subseteq B$ means every element of $A$ is an element of $B$; then $A$ is a subset of $B$ and $B$ is a superset of $A$.
• $A\subseteq B$ and $B\subseteq A$ together force $A=B$.
• For any set $A$: $\varnothing\subseteq A$ and $A\subseteq A$ -- these are the trivial subsets of $A$. $A\subseteq A$ in particular makes $A$ its own improper subset.
• $A$ is a proper subset of $B$ ($A\subsetneq B$) if $A\subseteq B$ and $A\ne B$: every element of $A$ is in $B$, and $B$ has at least one extra element.
• The standard chain of number systems is $N\subset W\subset Z\subset Q\subset R$, where $N$ = natural numbers, $W$ = non-negative integers, $Z$ = integers, $Q$ = rationals, $R$ = reals. The irrationals are a subset of $R$ but of none of $N,W,Z,Q$.

Union and intersection.

$A$ and $B$ are disjoint if $A\cap B=\varnothing$.

Indexed union/intersection. Just as $\sum_{i=1}^na_i$ abbreviates $a_1+a_2+\cdots+a_n$, we write

for $A_1\cup A_2\cup\cdots\cup A_n$ and $A_1\cap A_2\cap\cdots\cap A_n$ respectively.

Power set. For a set $A$, the power set $P(A)=\{B:B\subseteq A\}$ is the set of all subsets of $A$ (including $\varnothing$ and $A$ itself). If $n(A)=n$, then $n(P(A))=2^n$.

Universal set and complement. All sets in a given discussion are usually thought of as subsets of one fixed universal set $U$. If $A\subseteq U$, the complement of $A$ is $A'=A^c=\{x:x\in U\text{ and }x\notin A\}$.

Set difference and symmetric difference.

Immediate consequences: $U-A=A'$; $A-A=\varnothing$; $\varnothing-A=\varnothing$; $A-\varnothing=A$; $A-U=\varnothing$. …