Set Operations

A precise guide to union, intersection, difference, complement, Cartesian product, and set identities.

Set operations select regions according to membership conditions. Two overlapping sets sit inside a universal set. Distinct shading marks intersection, union-only regions, and complement so logical words correspond to visible regions.

Precise definition

For sets $A$ and $B$, the union $A\cup B$ contains elements in at least one set; the intersection $A\cap B$ contains elements in both; the difference $A\setminus B$ contains elements in $A$ but not $B$. Relative to a universe $U$, the complement is $A^c=U\setminus A$.

Notation and mathematical language

Membership notation $x\in A$ differs from subset notation $A\subseteq B$. The Cartesian product $A\times B=\{(a,b):a\in A,b\in B\}$ uses ordered pairs, so generally $A\times B\ne B\times A$. De Morgan's laws are $(A\cup B)^c=A^c\cap B^c$ and $(A\cap B)^c=A^c\cup B^c$.

Conceptual picture

Venn diagrams encode membership regions, but element-wise logic provides the proof. For example, $x\in A\cap(B\cup C)$ means $x\in A$ and ($x\in B$ or $x\in C$), which distributes to membership in $(A\cap B)\cup(A\cap C)$.

Fully worked example

Interpretation and application

Set operations express database filters, probability events, solution sets, and access-control groups. 'Not in A or B' is ambiguous in ordinary language; symbolic parentheses distinguish outside the union from membership in a complement combined with another set.