1. **Problem Statement:** Given a set defined by an extension of set-builder notation, such as $A = \{3x \mid x \in \mathbb{N} \text{ and } x < 5\}$, we want to write the set in roster form, determine its cardinality, find the number of subsets, and analyze if two sets are equal, equivalent, both, or neither. 2. **Understanding the notation:** - The expression before the vertical bar (here $3x$) defines the elements of the set. - The condition after the vertical bar restricts the values of $x$. - $\mathbb{N}$ denotes natural numbers (1, 2, 3, ...). 3. **Example given:** - $x$ takes values $1, 2, 3, 4$ (since $x < 5$). - Substitute into $3x$ to get elements: $3, 6, 9, 12$. - Roster form: $A = \{3, 6, 9, 12\}$. 4. **Cardinality:** - The cardinality $|A|$ is the number of elements in $A$. - Here, $|A| = 4$. 5. **Number of subsets:** - The number of subsets of a set with $n$ elements is $2^n$. - For $A$, number of subsets = $2^4 = 16$. 6. **Equality and equivalence of sets:** - Two sets are **equal** if they have exactly the same elements. - Two sets are **equivalent** if they have the same cardinality (number of elements), regardless of what the elements are. - They can be both equal and equivalent if they have the same elements. - They can be neither if they differ in elements and cardinality. **Summary:** - Write sets in roster form by substituting allowed values. - Count elements for cardinality. - Calculate subsets as $2^{\text{cardinality}}$. - Compare sets for equality and equivalence based on elements and cardinality. This method applies to any set defined by extension of set-builder notation.