Question 8.1.1: Unions and Intersections of Parts of a Line Using line AD, d......
Unions and Intersections of Parts of a Line
Using line AD, determine the solution to each part.
a) \vec{A B} ∩ \vec{D C} b) \vec{A B} ∪ \vec{D C} c) \overline{AB} ∩ \vec{CD} d) \overline{AD} ∪ \overset{\circ\rightarrow}{CA}
a) \vec{AB} ∩ \vec{DC}
Ray AB and ray DC are shown below. The intersection of these two rays is that part of line AD that is a part of both ray AB and ray DC. The intersection of ray AB and ray DC is line segment AD, \overline{AD}.
\vec{AB} ∩ \vec{DC} = \overline{AD}
b) \vec{AB} ∪ \vec{DC}
Once again ray AB and ray DC are shown below. The union of these two rays is that part of line AD that is part of either ray AB or ray DC. The union of ray AB and ray DC is the entire line AD, \overleftrightarrow{A D},
\vec{AB} ∪ \vec{DC} = \overleftrightarrow{AD}
c) \overline{AB} ∩ \vec{DC}
Line segment AB and ray CD have no points in common, so their intersection is empty.
\overline{AB} ∩ \vec{DC} = \varnothing
d) \overline{AD} ∪ \overset{\circ\rightarrow}{CA}
The union of line segment AD and half line CA is ray DA, \vec{DA} (or, equivalently, \vec{DB} or \vec{DC} ).
\overline{AD} ∪ \overset{\circ\rightarrow}{CA} =\vec{DA}