1. **Problem:** Find the union $X \cup Y$ where $X = (-7, 13)$ and $Y = (2, 10]$. 2. **Formula and rules:** The union of two sets $A$ and $B$, denoted $A \cup B$, is the set of all elements that are in $A$, or in $B$, or in both. 3. **Work:** Since $X = (-7, 13)$ includes all numbers between $-7$ and $13$ (not including endpoints) and $Y = (2, 10]$ includes numbers from $2$ to $10$ including $10$, the union covers from $-7$ to $13$. 4. **Answer:** $$X \cup Y = (-7, 13)$$ 1. **Problem:** Find the intersection $Y \cap X$. 2. **Formula and rules:** The intersection of two sets $A$ and $B$, denoted $A \cap B$, is the set of all elements that are in both $A$ and $B$. 3. **Work:** The overlap between $X = (-7, 13)$ and $Y = (2, 10]$ is from $2$ to $10$ including $10$ because $X$ includes all numbers up to but not including $13$. 4. **Answer:** $$Y \cap X = (2, 10]$$ 1. **Problem:** Find the complement $X'$ of $X$ relative to the universal set $U = (-\infty, +\infty)$. 2. **Formula and rules:** The complement of a set $A$ relative to $U$ is $A' = U \setminus A$, all elements in $U$ not in $A$. 3. **Work:** Since $X = (-7, 13)$, its complement is all real numbers less than or equal to $-7$ and greater than or equal to $13$. 4. **Answer:** $$X' = (-\infty, -7] \cup [13, +\infty)$$