Question 1141176: I just watched a video with the following explanation:
To Show that a set is countable:
• Formally, we have to show that there exists a bijective function between the set in question and the set of natural numbers.
• Informally, we have to show to the elements of the set can be put in an order, so that:
o No element will ever be repeated.
o And no element will ever be missed out.
I then watched another video that said that such are "one-to-One" and that such should be able to pass the "horizontal line test."
But X as 1,2,3,4,5 has no repeats, and Y as 6,9,7,11,8 also has no repeats.
And no numbers are left out.
And yet a horizontal line will pass through the graph twice, Ie, between 6 and 9 is 7.
I must be thinking about this wrong. Or is it correct to say that it is not the graph LINE that matters with the horizontal line test, it is just the actual points on that graph that must not be intersected by a horizontal line? ?
Found 2 solutions by Boreal, Edwin McCravy:
Answer by Boreal(15235)   (Show Source):
You can put this solution on YOUR website! That is correct. If these were points on a curve, it would not be 1:1, but as individual points, as long as one of them does not share two values of the other (a single y can't be defined by 2 different x values) then it is 1:1.
Answer by Edwin McCravy(20060)  (Show Source):
You can put this solution on YOUR website! X as 1,2,3,4,5 has no repeats, and Y as 6,9,7,11,8 also has no repeats.
So that means that the relation is
X 1 | 2 | 3 | 4 | 5 |
Y 6 | 9 | 7 |11 | 8 |
This is the set of ordered pairs:
{ (1,6), (2,9), (3,7), (4,11), (5,8) }
And so the graph of the relation looks like this:
When you plot them, you don't get a line or a curve, but just the 5 isolated
points plotted above.
You spoke of "a graph LINE". But there is no graph LINE at all.
Maybe you thought that the points had to be connected with LINES or maybe a
curved line like one of these:
 
But neither of those two graphs is the graph of the given relation. That's why
I X-ed them out. The points are NOT connected at all, but are separate points.
What you wrote:
"And yet a horizontal line will pass through the graph twice, Ie, between 6
and 9 is 7."
is false. That's because no horizontal line will pass through more than
ONE of the 5 points. That's why it's one-to-one. Look at these green horizontal
lines below:
Not a one of those horizontal green lines goes through more than ONE point. Some
horizontal lines go through 1 point and some go through NO points. But no
horizontal line can pass through more than one of the points of the relation. So
the relation is one-to-one.
Edwin
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