Introduction to Geometry (Level 5) In this video we are going to go over basic
examples that ask us to find the intersection and union of various geometric figures. Before
we jump into some examples lets refresh our memory and talk about sets. Say I have two
distinct sets let’s call them Set A and Set B, recall that a set is a collection of
objects, these objects are referred to as the elements of the set. In this example the
numbers 1, 2, 3, 4, 5, and 6 are the elements of set A, likewise the numbers 2, 4, 6, 8,10, and 12 are
the elements of set B. When dealing with sets we are often asked to find the union or intersection
between two or more sets. If we are asked to find the union between
set A and B we would denote it as follows: A union B, this symbol is pretty easy to remember
since it resembles the letter U for union. In essence, the union between two sets consists
of: {all elements in A or in B or in both}. You can think of a union as the process of
merging the elements of both sets together. In this example A union B will be equal to
a new set that contains every element that is in either set A or set B including those elements
that are common to both set A and B. In contrast, if we are asked to find the intersection
between set A and B we would denote it as follows: A intersection B. This symbol looks
like an upside down U and it sort of looks like the capital letter A if you use your
imagination, you can imagine the capital letter A will remind you of the word “and” since
the intersection of two sets creates a new set containing the elements in both A and
B. In this example A intersected B would equal to the set containing the elements {2,4 and 6}.
Since these elements are in both A and B, graphically you can see the reason why
this is called the intersection between two sets its literally an intersection or overlapping
of common elements. The final concept we need to be familiar with
is the idea of an empty set, let’s take Set A and Set B and remove the elements common to both sets.
Now if we are asked to find A intersection B we would see that there are no common elements
between set A and B, in this case we would obtain what is referred to as an empty set,
and we use this symbol to denote the empty set,
an empty set is special in the sense that it contains no elements.
Alright now that we are refreshed and familiar with the concepts associated with sets lets
go ahead and try the first example. Use the following figure to answer the following
questions. How many lines are shown? Name these lines.
By looking at the figure we can see that there are two distinct lines. So the answer to the
first question is 2. Now the second question is asking us to name them, let’s focus on
the line that contains points D, B and E, this line can be named by using two points
so we can name this line as line DE or line DB or line BD or line or line BE or line EB or line ED,
notice that this line is labeled
with the lower case letter m, so we can also name this line as line m. In the same manner,
the line that contains points A, B and C can be name as line AB or line AC or line BA or line BC or line CB or line CA.
Remember lines can be named by denoting two points located on the line or by using a lower
case letter whenever possible. Alright, let’s try the second example.
Where do line AC and line DE intersect? Using set notation this problem translates
to the following expression: line AC intersection line DE.
Looking at the lines we see that they intersect at a common
point in this case point B. So line AC intersection line DE is equal to point B.
Let’s try the next example.
Where does ray AC intersect line BC? Using set notation this problem translates
to the following expression. Recall that you can think of a ray as being the set of all
points that starts at one end point in this case point A and goes in the direction of
another point in this case point C, in the same manner a line is the set of all points
that extends infinitely in both directions so the following points are located on line
BC, for this problems we are looking for the points that both ray AC and line BC have in
common hence the intersection, in this case the common points start intersecting at point
A and extend infinitely in the direction of point C, therefore the intersection of ray
AC and line BC is equal to ray AC. Alright let’s try the next example.
What is the union of ray BA and ray BD? Using set notation this problem translates
to the following expression. First let’s locate each of the rays, ray BA is located
here and ray BD is located here, you can think of the union as the merger between these two rays,
we want to include all the points on ray BA or BD or in both. Recall that an angle is
formed when two rays have a common point of intersection called the vertex this is essentially
what happens when we form the union of these two rays, therefore the union of ray BA and
ray BD is equal to angle ABD or angle DBA. Alright let’s move along to the next example.
Find the intersection of ray AB with ray CA. Using set notation this problem translates
to the following expression. Once again let's identify the set of all points located on ray
AB, then we need to identify the set of all points on ray CA, now it’s just matter of finding
where the set of points of these two rays intersect, we see that the set of points on
both rays start intersecting at point A and end at point C, in essence the set of points
intersect along a common line segment in this case line segment AC. So the intersection
of these two rays is equal to line segment AC. Alright, let's end the video with the final example.
Find the intersection of triangle ABC and ray DE.
Using set notation this problem translates to the following expression. The set of all
points located on triangle ABC are represented as follows, and the set of all points located
on ray DE are represented as follows, for this problem we are looking for the intersection
of these two set of points. Notice that both sets intersect at a common point in this case
they intersect at point F, so the intersection of triangle ABC and ray DE is the set containing
point F. Alright in our next video we will continue
covering slightly more challenging examples involving the union and intersection of geometric figures.