Sets: union, intersection, complement

A set is a collection of objects, called elements. We write $x \in E$ to say that $x$ belongs to $E$, and $x \notin E$ otherwise.

Sets are one of the basic languages of modern mathematics, and they are closely tied to logic: each operation on sets corresponds exactly to a logical connective on membership propositions.

The union $A \cup B$ gathers all the elements belonging to $A$ or to $B$, in the inclusive sense. The intersection $A \cap B$ keeps only the common elements, belonging to $A$ and to $B$. The complement of $A$ in a reference set $E$, written $E \setminus A$, gathers all the elements of $E$ that do not belong to $A$: it is the set-theoretic translation of negation.

Set operation	Corresponding logical connective
$A \cup B$	or, $\vee$
$A \cap B$	and, $\wedge$
$E \setminus A$	not, $\neg$

These correspondences are not mere decorative analogies: they allow a proof about sets to be translated into a purely logical proof, and vice versa.

Thus, De Morgan's laws state that the complement of a union is the intersection of the complements:

$$(A \cup B)^c = A^c \cap B^c \qquad (A \cap B)^c = A^c \cup B^c$$

exactly as the negation of an "or" is the "and" of the negations.

Reading a Venn diagram

The Venn diagram is the standard visual tool: two overlapping circles represent two sets, and each region of the diagram corresponds to a specific combination of membership.

Region of the diagram	Corresponding set
the central zone, shared by both circles	$A \cap B$
both circles together, central zone included	$A \cup B$
circle $A$ minus the central zone	$A \setminus B$
everything outside circle $A$	the complement of $A$ in $E$

Learn to read these regions fluently: they help you visualize a property instantly, even before trying to prove it formally.