Venn Diagrams

Fig. 1.2 is the simplest possible Venn diagram — a single circle inside a rectangle. The rectangle is labelled $U$ and represents the universal set, which in this case is $\{1, 2, 3, \dots, 10\}$. The circle inside it is labelled $A$ and represents a subset of $U$. The elements of $U$ are shown as dotted points placed either inside the circle or outside it, depending on whether they belong to $A$ or not.

The points $2, 4, 6, 8, 10$ lie inside the circle $A$, so $A = \{2, 4, 6, 8, 10\}$. The points $1, 3, 5, 7, 9$ lie inside the rectangle but outside the circle, meaning they belong to $U$ but not to $A$. The diagram therefore shows visually that $A$ is a subset of $U$, written $A \subset U$, and that $A$ contains exactly the even numbers from $1$ to $10$.

The physical idea is simple but powerful: a Venn diagram turns a set relationship into a spatial picture. The rectangle gives the "universe" of all possible elements, and each closed curve (here a circle) carves out the region belonging to a particular set. An element's membership is decided by which region it falls into — inside the circle means "in $A$", outside the circle but inside the rectangle means "in $U$ but not in $A$". This is the foundation for representing union, intersection, and difference later in the chapter.

The key formula that this figure illustrates is the definition of a subset:

Fig 1.3 is a Venn diagram that shows the relationship of subset between two sets. The universal set $U$ is the rectangle containing everything. Inside it, two circles are drawn: a larger circle labelled $A$ and, completely inside $A$, a smaller circle labelled $B$. The fact that $B$ lies entirely within $A$ is the visual representation of $B \subset A$ — every element of $B$ is also an element of $A$.

The elements of $U$ are placed as points inside the appropriate regions. The numbers $4$ and $6$ sit inside the inner circle $B$ (and therefore also inside $A$). The numbers $2$, $8$, and $10$ lie inside $A$ but outside $B$ — they belong to $A$ but not to $B$. The numbers $1$, $3$, $5$, $7$, $9$ are placed outside both circles but still inside the rectangle $U$; they belong to the universal set but to neither $A$ nor $B$.

So the diagram tells you, at a glance:

• $U = \{1,2,3,4,5,6,7,8,9,10\}$
• $A = \{2,4,6,8,10\}$
• $B = \{4,6\}$
• $B \subset A$ (the inner circle is completely inside the outer one)

The textbook uses this figure to prepare you for the operations of union, intersection, and difference. Once you see that $B$ sits inside $A$, you can immediately read off:

• The intersection $A \cap B$ is $\{4,6\}$ — the region where the two circles overlap (here, the whole of $B$).
• The difference $A - B$ (or $A \setminus B$) is $\{2,8,10\}$ — the part of $A$ outside $B$.
• The union $A \cup B$ is just $A$ itself, $\{2,4,6,8,10\}$, because $B$ adds no new elements.