1. Let's start with the **definition of a set**: A set is a collection of distinct objects, considered as an object in its own right. 2. Sets are usually denoted by capital letters such as $A$, $B$, and $C$. 3. The objects inside a set are called elements or members, and we write $a \in A$ to denote that element $a$ is in set $A$. 4. **Basic operations on sets**: - **Union ($A \cup B$)**: The set of all elements that are in $A$ or in $B$ or in both. - **Intersection ($A \cap B$)**: The set of all elements that are in both $A$ and $B$. - **Difference ($A - B$)**: The set of elements that are in $A$ but not in $B$. - **Complement ($A^c$)**: The set of all elements not in $A$ (usually relative to a universal set $U$). 5. **Special sets**: - The **empty set** $\emptyset$ has no elements. - The **universal set** $U$ contains all possible elements under consideration. 6. **Set equality**: Two sets $A$ and $B$ are equal if every element of $A$ is in $B$ and every element of $B$ is in $A$. 7. **Subset**: $A$ is a subset of $B$ (written $A \subseteq B$) if every element in $A$ is also in $B$. 8. Let's illustrate with an example: Let $A = \{1, 2, 3\}$ and $B = \{2, 3, 4, 5\}$. 9. Then, - $A \cup B = \{1, 2, 3, 4, 5\}$ - $A \cap B = \{2, 3\}$ - $A - B = \{1\}$ 10. We have covered the fundamental concepts of set theory from scratch.