1. The problem is to explain the difference between a group and a groupoid. 2. A **group** is a set $G$ equipped with a binary operation $\cdot$ satisfying four properties: - **Closure**: For all $a, b \in G$, $a \cdot b \in G$. - **Associativity**: For all $a, b, c \in G$, $(a \cdot b) \cdot c = a \cdot (b \cdot c)$. - **Identity element**: There exists an element $e \in G$ such that for all $a \in G$, $e \cdot a = a \cdot e = a$. - **Inverse element**: For each $a \in G$, there exists $a^{-1} \in G$ such that $a \cdot a^{-1} = a^{-1} \cdot a = e$. 3. A **groupoid** generalizes the concept of a group by allowing partial operations: - A groupoid is a category where every morphism is invertible. - Informally, it consists of a set of elements for which the binary operation (composition) is only partially defined. - It has a set of objects, and elements (morphisms) between these objects that can be composed only when the source of one matches the target of another. - Every morphism has an inverse. 4. In summary: - A group is a groupoid with exactly one object where the operation is defined everywhere. - A groupoid can have many objects and partial operations, while a group has a single set and a total operation. Therefore, **every group is a groupoid, but not every groupoid is a group**. This completes the explanation.