1. Let's start by stating the problem: understanding the difference between elements and subsets in set theory. 2. An **element** is a single object or member contained within a set. For example, if we have a set $A = \{1, 2, 3\}$, then $1$ is an element of $A$, written as $1 \in A$. 3. A **subset** is a set whose elements are all contained within another set. For example, $B = \{1, 2\}$ is a subset of $A$, written as $B \subseteq A$. 4. Important rules: - If $x$ is an element of $A$, we write $x \in A$. - If every element of $B$ is also in $A$, then $B$ is a subset of $A$, written $B \subseteq A$. - Every set is a subset of itself. 5. To summarize: - Elements are individual objects inside a set. - Subsets are sets made up of elements from another set. 6. Example: Given $A = \{1, 2, 3\}$: - $2$ is an element of $A$ ($2 \in A$). - $\{1, 3\}$ is a subset of $A$ ($\{1, 3\} \subseteq A$). This distinction helps us understand how sets and their members relate in mathematics.