Q.In the set $\mathbf{Z}$ of integers, define $mRn$ if $m-n$ is divisible by 7. Prove that R is an equivalence relation.

A set is a well-defined, distinguishable collection of objects -- given any object, we must be able to decide for certain whether it belongs. "Beautiful flowers" is not well-defined (beauty is subjective); "red flowers in a named garden" is.

Subsets. $A\subseteq B$ means every element of $A$ lies in $B$. Mutual inclusion ($A\subseteq B$ and $B\subseteq A$) forces $A=B$. For any $A$: $\varnothing\subseteq A$ and $A\subseteq A$ are its trivial subsets (the second makes $A$ its own improper subset); $A\subsetneq B$ ("proper subset") additionally requires $A\ne B$, i.e. $B$ has at least one extra element. The number-system chain is $N\subset W\subset Z\subset Q\subset R$.

A set can even be an element of another set: if $A=\{1,2\}$ and $B=\{1,\{1,2\},3,4\}$, then $A\in B$, since the single object $\{1,2\}$ is literally listed as one of $B$'s four members. (This does not automatically make $A\subseteq B$ -- here it does not, since $2\notin B$ as an individual element.)

Union, intersection, complement, difference. For a fixed universal set $U$:

$A,B$ are disjoint if $A\cap B=\varnothing$. Indexed forms $\bigcup_{i=1}^nA_i,\ \bigcap_{i=1}^nA_i$ extend union/intersection to many sets at once.

Power set. $P(A)=\{B:B\subseteq A\}$, the set of all subsets of $A$; if $n(A)=n$, then $n(P(A))=2^n$ (each of the $n$ elements is independently in or out of a candidate subset). …