Q.Let $U$ be the set of all boys and girls in a school, $G$ be the set of all girls in the school, $B$ be the set of all boys in the school, and $S$ be the set of all students in the school who take swimming. Some, but not all, students in the school take swimming. Draw a Venn diagram showing one of the possible interrelationship among sets $U$, $G$, $B$ and $S$.

Concept: Set relationships and Venn diagrams with partition and overlap.

The universal set $U$ contains all boys and girls in the school. Since every student is either a boy or a girl (and not both), $G$ and $B$ partition $U$: they are disjoint and their union is $U$.

The set $S$ (students who swim) cuts across both $G$ and $B$ because some (but not all) students swim. This means $S$ overlaps with both $G$ and $B$, but neither contains the other entirely.

A valid Venn diagram:

• Draw a rectangle for $U$
• Inside, draw two non-overlapping regions for $G$ and $B$ that together fill $U$
• Draw a circle for $S$ that intersects both $G$ and $B$, leaving parts of each outside $S$ …

Understanding the sets and their relationships

The problem gives us four sets with natural constraints. $U$ is the universal set containing every student. Since every student is either a girl or a boy (and not both), $G$ and $B$ partition $U$: they are disjoint and their union is $U$. In symbols, $G \cap B = \emptyset$ and $G \cup B = U$.

The set $S$ represents students who swim. The phrase "some, but not all" tells us two things: at least one student swims ($S \neq \emptyset$), and at least one student does not swim ($S \neq U$). Crucially, nothing restricts swimmers to one gender—both girls and boys can swim. So $S$ can overlap with both $G$ and $B$.

Constructing the Venn diagram

1. Draw the universal set $U$.

   Start with a rectangle representing all students in the school.

2. Partition $U$ into $G$ and $B$.

   Inside the rectangle, draw two non-overlapping regions (circles or ovals) that together fill $U$. Label one $G$ (girls) and the other $B$ (boys). These must not intersect because no student is both a girl and a boy.

3. Add the set $S$ cutting across both $G$ and $B$.

   Draw a third region (another circle or oval) that overlaps with both $G$ and $B$, but does not cover either completely. Label it $S$. This shows:

   • Some girls swim: $S \cap G \neq \emptyset$
   • Some boys swim: $S \cap B \neq \emptyset$
   • Some girls don't swim: there's a part of $G$ outside $S$
   • Some boys don't swim: there's a part of $B$ outside $S$ …