Hello. I'm Professor Von Schmohawk and welcome to Why U.
We have seen how ordered pairs can be graphically represented as points in a plane
using Cartesian coordinates.
As we will see later in this lecture
this set of ordered pairs is an example of a "binary relation".
A binary relation relates various elements of one set to elements of another set.
For example, let’s say that set A contains Tarzan, Jane, and Boy
and set B contains oranges, apples, and bananas.
Now, how might the elements of these two sets be related?
One possible relation between these people and fruit
would be that certain people in set A like certain fruit in set B.
For instance, let's say that Tarzan likes oranges
Boy likes apples and bananas
and Jane likes oranges, apples, and bananas.
Instead of drawing multiple arrows from a single person
we could represent this relation by a collection of individual arrows
each connecting one person to one fruit.
Or instead of drawing arrows
we could represent each association between a person and a fruit as an ordered pair.
In each ordered pair, the first element is a person from set A
and the second element is one fruit from set B which that person likes.
We call these pairs of elements "ordered"
since the two elements cannot be swapped without changing the meaning of the relation.
For example, Tarzan likes oranges
but oranges don't necessarily like Tarzan.
This collection of ordered pairs symbolizes associations
from members of one set to members of another set.
We call the set that contains these ordered pairs a "binary relation".
Let's name this set of ordered pairs L.
So we say that L is a "binary relation from set A to set B".
Each ordered pair that's a member of set L
makes a statement about how one person in set A is related to one fruit in set B.
A mathematician would say that this ordered pair makes the statement
"Jane is L-related to oranges"
or in plain English, "Jane likes oranges".
You may recall from previous lectures
that the Cartesian product of two sets is also a set of ordered pairs.
Unlike relation L, the Cartesian product contains every possible ordered pair which can be created
where the first element of the ordered pair is a member of set A
and the second element is a member of B.
The Cartesian product of sets A and B therefore contains all nine possible ordered pairs
as opposed to the six contained in relation L.
Of course, everyone in set A could have liked every fruit in set B.
In that case, relation L would have been the same as the Cartesian product.
On the other hand, it might be that nobody in set A likes fruit.
In that case relation L would be the empty set.
So a binary relation from one set to another is always a subset of their Cartesian product
since it can contain as many as all
or as few as none of the ordered pairs in the Cartesian product.
Binary relations don't necessarily have to involve two different sets.
A binary relation can exist between members of the same set.
For example, we could take two copies of set A which contains Tarzan, Jane, and Boy
and draw arrows from each member to every member who is shorter.
So this relation is represented by three ordered pairs
where the first element of each ordered pair is the taller person
and the second element is the shorter person.
Let's call this relation T.
Since both sets involved in this binary relation are the same set, A
we call T a binary relation "on" set A.
Each ordered pair in set T
makes a statement about how one person in set A
is related to a person in the same set.
This ordered pair makes the statement "Tarzan is T-related to Jane"
or in plain English "Tarzan is taller than Jane".
As we saw, a binary relation from one set to another
is a subset of their Cartesian product
so T is a subset of the Cartesian product of set A with itself
or A squared.
A squared contains all nine possible ordered pairs
which can be created from the three members of set A.
If we eliminate all the ordered pairs of A squared
whose first element is not a person taller than the second element
we get relation T.
In the beginning of this lecture, we showed a group of points in the xy plane
and said that this is an example of a binary relation.
So what are the two sets in this binary relation and how are they related?
As we saw, a point on the xy plane
is a visual representation of an ordered pair of real numbers
where the first element of the ordered pair corresponds to a number on the x axis
and the second element corresponds to a number on the y axis.
So the ordered pair relates
one member of the set of real numbers represented by the x axis
to one member of the set of real numbers represented by the y axis.
A set consisting of this ordered pair
would thus be a binary relation from one set of real numbers to the other
and the corresponding point on the xy plane is a visual representation of this relation.
Of course, a binary relation can include more than one ordered pair.
So the group of points we showed in the beginning
is a visual representation of a binary relation on R, the set of real numbers.
As we saw, a binary relation between two sets is a subset of their Cartesian product.
So this binary relation
is a subset of the Cartesian product of the set of real numbers with itself, R-two
which consists of every point in the xy plane.
In the next lecture, we will introduce two important sets in any binary relation
called the "domain" and the "range" of the relation.