1. **Stating the problem:** We want to understand the different types of sets in mathematics, their definitions, and examples. 2. **Definition and types of sets:** - A **set** is a collection of distinct objects, called elements. - Types of sets include: - **Empty set (Null set):** Contains no elements. Example: $\emptyset = \{\}$. - **Finite set:** Has a countable number of elements. Example: $A = \{1,2,3\}$. - **Infinite set:** Has unlimited elements. Example: $\mathbb{N} = \{1,2,3,\ldots\}$. - **Subset:** Set $A$ is a subset of $B$ if every element of $A$ is in $B$. Notation: $A \subseteq B$. - **Proper subset:** $A$ is a subset of $B$ but $A \neq B$. Notation: $A \subset B$. - **Universal set:** The set containing all elements under consideration, denoted $U$. - **Power set:** The set of all subsets of a set $A$, denoted $\mathcal{P}(A)$. - **Disjoint sets:** Sets with no common elements. 3. **Examples and workings:** - Empty set: $\emptyset = \{\}$ has no elements. - Finite set: $A = \{2,4,6\}$ has 3 elements. - Infinite set: $\mathbb{Z} = \{\ldots,-2,-1,0,1,2,\ldots\}$. - Subset example: If $A = \{1,2\}$ and $B = \{1,2,3\}$, then $A \subseteq B$. - Power set example: For $A = \{1,2\}$, $\mathcal{P}(A) = \{\emptyset, \{1\}, \{2\}, \{1,2\}\}$. - Disjoint sets example: $A = \{1,2\}$ and $B = \{3,4\}$ have no common elements. 4. **Important rules:** - Elements in a set are unique; duplicates are not counted. - Order of elements does not matter: $\{1,2\} = \{2,1\}$. - To check subset, verify every element of $A$ is in $B$. 5. **Summary:** Sets are fundamental in math to group objects. Knowing types helps in understanding relations and operations on sets.