Algebra 13 – Domain and Range of Binary Relations
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Algebra 13 – Domain and Range of Binary Relations


Hello. I’m Professor Von Schmohawk
and welcome to Why U. In the previous lecture we saw how to create
a binary relation between two sets. These sets can contain any type of elements. We showed that a binary relation is a way of associating or relating elements
of one set to elements of the other set. For instance, we could draw arrows
from each person in set A who likes fruit to the fruits in set B which they like. As we saw, this same information can be conveyed
using ordered pairs where the first element of each ordered pair
is a member of the first set and the second element
is a member of the second set. Let’s name a set containing these
ordered pairs, set L. So L represents a binary relation
from A to B. In a binary relation not every element in the two sets which are
related may be involved in the relation. In our example, Tarzan, Boy, oranges, apples,
and bananas are part of relation L while Jane, pears, and cherries are not. The set of all elements of A
which are L-related to elements of B is called the “domain” of L. Likewise, the set of all elements of B
to which the elements of A are related is called the “range” of L. So in a binary relation from A to B all the elements of A which are related
form the domain and all the elements of B which are related
form the range. The domain and range
are subsets of A and B. The domain and the range can include
every element of A and B or could include just a single element. We have illustrated binary relations, between
sets containing various types of elements. But often in Algebra
the sets which are related are both equal to the
set of real numbers R. In this case, we can either say that
the binary relation is from R to R or more simply
that the binary relation is “on” R. The ordered pairs which represent a binary
relation on R are pairs of real numbers. This set of ordered pairs is sometimes referred
to as the “graph” of the relation. And since Cartesian coordinates can be used to visually represent ordered pairs
of real numbers this is a way to visually represent
the graph of a relation on R. Creating this visual representation of a relation
is called “graphing” the relation and is one of the most powerful methods
for understanding mathematical relations. There are many types of graphs. Graphs of finite sets of ordered pairs
like this are called “scatter plots”. In the next lecture, we will see how
scatter plots can be an important tool allowing us to visually recognize patterns
which may exist in a binary relation.

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