Venn Diagrams — Picturing Set Operations

A Venn diagram represents sets as overlapping regions, usually drawn as circles (or ovals) inside a rectangle representing the universal set $U$. Every set operation from the last two sections has a direct, visual counterpart in a Venn diagram — this is exactly why Venn diagrams are the standard tool for verifying set-operation properties without needing an algebraic proof, which is precisely how this syllabus uses them.

The figure above places every element of $A = \{2,4,6,8,10\}$ and $B = \{4,8,12,16\}$ (the same pair used in Section 3) into its correct region: $2, 6, 10$ inside $A$ only; $12, 16$ inside $B$ only; and $4, 8$ inside the overlapping lens shared by both. Once every element is placed like this, each operation can be read off directly from the same picture: $A \cup B$ is every element shown; $A \cap B$ is just the lens, $\{4,8\}$; and $A - B$ is the part of $A$ outside the lens, $\{2,6,10\}$.

Reading a Venn diagram is a matter of matching the shaded region to its defining condition: $A \cup B$ shades everywhere that is in $A$ or $B$ (both circles, no gaps); $A \cap B$ shades only the lens-shaped overlap (in both at once); $A - B$ shades only the part of $A$ that is outside $B$; and $A'$ shades everything in the surrounding rectangle $U$ that falls outside the circle for $A$. …