Union Intersection Sets 056A3E

1. **Problem Statement:**\nWe are given rays $\overrightarrow{AB}$ and $\overrightarrow{BA}$ and segment $\overline{AB}$. We need to find the union and intersection of these sets of points as described in parts a to f.\n\n2. **Definitions and Important Rules:**\n- A ray $\overrightarrow{AB}$ starts at point $A$ and passes through $B$, continuing infinitely beyond $B$.\n- A ray $\overrightarrow{BA}$ starts at point $B$ and passes through $A$, continuing infinitely beyond $A$.\n- A segment $\overline{AB}$ consists of all points between $A$ and $B$, including $A$ and $B$.\n- The union $X \cup Y$ of two sets $X$ and $Y$ is the set of points in $X$ or $Y$ or both.\n- The intersection $X \cap Y$ of two sets $X$ and $Y$ is the set of points common to both $X$ and $Y$.\n\n3. **Analyze each part:**\n\na. $\overrightarrow{AB} \cup \overrightarrow{BA}$\n- $\overrightarrow{AB}$ is ray starting at $A$ through $B$ and beyond.\n- $\overrightarrow{BA}$ is ray starting at $B$ through $A$ and beyond.\n- Their union covers the entire line through $A$ and $B$.\n\nAnswer: The entire line through $A$ and $B$.\n\nb. $\overrightarrow{AB} \cap \overrightarrow{BA}$\n- The rays start at different points and extend in opposite directions.\n- The only points common to both rays are the points on the segment $\overline{AB}$.\n\nAnswer: The segment $\overline{AB}$.\n\nc. $\overline{AB} \cup \overrightarrow{BA}$\n- $\overline{AB}$ is the segment between $A$ and $B$.\n- $\overrightarrow{BA}$ is the ray starting at $B$ going through $A$ and beyond.\n- Their union is the ray $\overrightarrow{BA}$ because the segment is contained in the ray.\n\nAnswer: The ray $\overrightarrow{BA}$.\n\nd. $\overline{AB} \cup \overrightarrow{AB}$\n- $\overline{AB}$ is the segment between $A$ and $B$.\n- $\overrightarrow{AB}$ is the ray starting at $A$ going through $B$ and beyond.\n- Their union is the ray $\overrightarrow{AB}$ because the segment is contained in the ray.\n\nAnswer: The ray $\overrightarrow{AB}$.\n\ne. $\overline{AB} \cap \overrightarrow{AB}$\n- The segment $\overline{AB}$ is contained entirely within the ray $\overrightarrow{AB}$.\n- So their intersection is the segment $\overline{AB}$.\n\nAnswer: The segment $\overline{AB}$.\n\nf. $\overrightarrow{AB} \cap \overrightarrow{BA}$\n- This is the same as part b.\n- Intersection is the segment $\overline{AB}$.\n\nAnswer: The segment $\overline{AB}$.