You know it want to be good. You know numbers
are just like you and I, They like to travel in pairs
and when you get a buncha pairs together,
well ,you kinda have a little party. You mind telling me what this is all about. mister.
Well, that's what a relation is. It's a group of pairs
or a set of ordered pairs.
Here are some examples. This we'll call relation A.
It's got the pairs (4,6)
and another pair (0,7)
(6,2) and (2,6). Now note that these are ordered pairs
so (6,2) is different then (2,6)
Here's another way to write a relation.
This is the relation B which is the set
of 3 ordered pairs. The pair (1,2),
(3,4) and (5,6). Starting to look a little like points, aren't they?
Here's another one. It is a relation.
Relation C, the set of (1,4),
(1,3), (1,2) and (1,6).
And yet another, relation D.
Same idea. A set of ordered pairs.
And don't forget what I told you.
Wait a minute. Order counts. These two pairs are different but we're gonna
get in the habit of listing X then Y. iI's kind of easy to do because they're
in alphabetical order.
Lookit here, I say lookit! To set up all the first
numbers or we'll call them the inputs in a relation
is called [Drumroll]
the domain.
aAl these red numbers are the domains, if you would,
of these various relations that I just showed you.
Hmmmm. Basically it's all the
Xs if you have points.
We also call that the independent variable. X is the independent variable
because you get to pick it.
Okay? You get to independently pick it. Now that's the first numbers
where you are but what Am I while the second numbers
or the outputs a function or a relation, if you would,
is called the range. All the second numbers
the yes although wise you're right
now with the first number you get to pick them
their independent lately but the
second numbers depend on what the formula is up the relation
now okay so that the pen it's going to be called the dependent variable
the wise are the dependent variable their value depends
on what the formula is to map from x to y
hell but let's stick with forget the formula for
for now let's stick with naming the domain and range
in this relation what is the domain
well it's the set of all the first numbers all the axis
a set one two three and four and
the range is all the second numbers 246 in a very simple isn't it
well let's make it a little bit more difficult
initial be a little in this case
the domain is the set all the first number so it's
the said 0123 now the range
is this set have all the second numbers but we never list
a number twice in a set so the range is simply the set
5 another way to say that is that it's all the possible
second numbers old okay only list each value
wants now let's
create a graph every relation
by throwing some points on the graph it remember a point is
I parer each point has an axe
but and the Y now what's the domain and range
I'll what we have pictured here the domain
is going to be all the X values well we can look at this
whom and the X is moving from left to right are gonna be
let's see rap
okay let's throw an axis in there the axes are going to be
-6 0
note I'm going from left to right from smallest to largest
to and
7 all now that was a domain
the range is going to be all I'm
the possible why so we're gonna go from left to right we're gonna go from
bottom to top he's getting little numbers on the axis there
and from bottom to top the whys are gonna be this Sat Nav
0 that's how high that is
what's the next one one
now be careful what's the next one moving up
who now moving from left to right are we
and five put all
1 and that is the domain and range in this picture
although it and its do another I'm
know a couple more points on their
little trickier
the domain remember whom is all the possible axes this Sat
I love all the possible access and this picture we have
no problem to
4
moving from left to right 6
an eight you now that's the domain the possible
axis
the range of course whom is the possible wise I should say this Sat Nav
all the possible wives gonna be a little smaller here is in it because the
possible wise
r5 what's else
5 you we don't say it again doing
5 kinda and
5 cool or not so that's the range
you just list 5 as some people like to say the domain
is how wide a picture is and the ranges
how tall a picture is and it will get an idea that when we look at this
really can list all the points let's start
but by naming the domain
which is all the possible access we can't list them all
do you but we know they go between these two numbers
and we're not going to include where we've circled hope you get the idea
so it somewhere
all the points between two and four not including two
and including 4 and we're going from left to right
the notation will use here but is gonna have to name
and infinite number of points isn't it excuse me
so we'll have in quality notation and his little fancy
this is red no a lot all the X's
and that line is such that all the axe is such that
X is you read the axe first now
X is greater than 2 and less than or equal to 4
note that we we're interested in all the possible access so we say the axe first
know what likelihood now to lou possible wise
EU we're going from lowest to highest
and in this case we're going from zero to six we're not going to include
0 but we are going to include six so we're talking about
but once again in infinite number of points
so we'll have to use any quality notation once again
its you need me to pick your brain is what it'll be
it'll be all the wise notice the range is why is now this time
all the Y's such that
member that mean such that I'm gonna read this with the white first all the
Y's such that why is greater than 0 and less than or equal to 6
and that is the range that's about the only way to do it remove the
removable okay let's review
inequality notation above
when we have a solid circle it means we include the endpoint
we're gonna do that when we have a greater than or equal to
and a less than or equal to home we haven't
open circle I'll means we don't want to include the end point to that would come
up when we have
unless then are greater than or equal to if you would
okay bowl right
let's try another and this green graph
is gonna have a domain defined by all the possible
axis and once again we're gonna go from left to right
our lowest axes 0
doing way our highest tax goes on for ever
so what are we gonna do here girl privacy i
well we're gonna have to use any quality notation again and it's going to be such
that
questioning axe the set of all axes
such that axes greater than or equal to 0 in that sense it on into infinity
room the wise our little easier this time
the wise just four and we could just list for
but you might also see something fancy like this which would be read
all the set of all wise such that why he calls for
don't let that scare you azo it will be
court kill a separate oh boy this about 1800
let's try this one hopefully can see once again that them
the domain measures kind of how wide something is
and the range measures how tall or short it is
the domain I'm is all the possible X is now this thing goes
to the left for ever and to the right for ever
the how we can name all those points
it or older notice the arrows say it goes on for ever in that direction
on what you can say is the set of all real numbers and we even have a symbol
for that only ID
if you see this are it means all real numbers
okay that's a possible domain otherwise
whom or all the possible where the rangers all the possible wise
it doesn't we go from lowest to highest it doesn't go down all the way the hell
does it
it starts at zero doing
better day and it goes up for ever
to it goes from zero to infinity how would we say that
we have to symbolize it with the Aeros
but to say it and set notation as the range
to will say all the Y's such that why is greater than a
or equal to 0 and that would be the range
possibilities baby okay go try doing that on your own
do the homework