WEBVTT

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Hi, my name is Richard.

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We're looking at Tamura 2021, paper two,
question 13.

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Let's read it through.

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A region R in the XY plane is defined by
the simultaneous inequalities y -,

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X is less than 3 and y -,
X ^2 is less than one.

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Which of the following statements is or
are true for every point in this region R?

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Statement 1 - 1 is less than X is less
than two.

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Statement 2 y -, X * y -,
X ^2 is less than three.

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Statement 3 Y is less than five.

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OK,
so I think the best approach to this is

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to sketch the region R and then look at
it and try and decide which of these

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three are true.

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So we have these two simultaneous
inequalities.

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I think they're slightly easier for us to
work with if we rearrange them so the

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first one is the same as Y is less than X
+ 3.

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If we want to imagine the set of points
which satisfy this whose X&

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Y coordinates satisfy this,
we might begin by thinking about the line

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y = X + 3.

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So the points on this line have their Y
coordinate equal to their X coordinate

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plus three.

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We want the Y coordinate to be less than
the X coordinate plus 3,

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and that's the points which are below
this line.

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If you want to convince yourself about
that,

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perhaps test the origin where Y is 0 and
X is 0, so 0 is certainly less than 0 + 3,

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so the origin satisfies this,
and from that,

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hopefully we can commit to ourselves.

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It's the points below this line that we
need.

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For the second inequality,
let's do the same thing.

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Let's rearrange this into Y is less than
X ^2 + 1.

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So again let's think about the set of
points where y = X ^2 + 1 first of all.

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So that will just be the graph of y = X
^2 but moved up by 1 unit.

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So that will look something like this,
bearing in mind that that was 3,

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so this will be 1.

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And once again,
we want the point to below that curve.

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Again, just test, test.

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The origin X IS0Y is 0,
so 0 is less than 0 ^2 + 1.

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The origin satisfies it,
and it's all the points below that curve.

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Now for R,
we want both of these to be true,

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so we need to be both below the line and
below the curve.

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And if we imagine what that looks like,
we would have this region here.

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Notice I'm carefully going to the curve
in this part and then back to the line

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here.

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And one thing to notice about R,
I've only got a part of it here,

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but it would extend indefinitely up in
this direction and down in this direction

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and down to.

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So that's worth considering as we start
to think about 1-2 and three.

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Let's start by thinking about #1 So is it
true that every point in R has it's X

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coordinate between -1 and 2?

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So that would mean that they were all
sort of in this region.

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But we can imagine there are points with
X coordinates as big as we like off to

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the right here.

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So for example, if you want to pick one,
there might be this point here 70.

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So X is 7 and Y is 0.

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It's in the region because 0 -,
7 is less than 3 and 0 -,

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7 ^2 is less than one.

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But you can see that 7 is not between -1
and 2.

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So certainly 1 is not true for every
point in R Likewise for three,

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we can imagine that we if we if we look
in this direction,

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we can find points with Y coordinate
greater than or equal to 5 S 3 will not

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be true either for all points in R We can
just take a point up here whose Y

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coordinate is bigger than five and see
that's not true.

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Number 2 is a little bit more difficult
to think about.

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If we look at the relationship between
two and the two inequalities we have,

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we can see that the left hand side of two
is the product of these two left hand

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sides and the right hand side is the
product of the two right hand side.

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So it might be easy to think that this
follows from these two inequalities.

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However, if we stop and think for a while,
is it true that if we have a value less

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than three and a second value less than
one,

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then their product will be less than 3 *
1?

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If you think about extreme cases of that,
perhaps a very large negative value which

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is less than three,
and another very large negative value

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which is less than one,
the product of those two would be a large

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positive value,
and it certainly wouldn't be less than 3.

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So we can see a kind of flaw in thinking
that this is true as a consequence of

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these two being true.

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So we need to interpret that in terms of
our diagram.

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I've already noticed that if this was a
large negative,

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if these were two large negative values,
then this would this bit of logic would

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not work.

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So what I'd like to get is these to be
large negative values.

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And if I take Y to be 0 and X to be a
large positive value.

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So for example,
coming back to my picture here further to

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the right, I might have a 10, 0.

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So if we look at this point, well, y -,
X is -10 Y is 0 and X is ten y -,

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X ^2 is -100 and the product of -10 and
-100 is +1000, which is not less than 3.

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So this point here is in our 10 zero.

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But it does not satisfy this.

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And we arrived at that point by thinking
about how the fact that this is true and

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this is true does not lead to this being
true.

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So it looks to me as though none of these
statements are true.

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And that means the answer we are looking
for is a here.

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One thing we might take away from this
question is that when we're working with

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inequalities and we're trying to decide
whether inequalities are a logical

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consequence of other inequalities,
it's sometimes useful to think about very

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extreme cases of those inequalities.

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So in this case, we,
we imagined that this y -,

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X was a big negative value,
and this was also a very big negative

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value.

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And we could see that the product there
will be a large positive value.

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It wouldn't be less than 3.

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So and that lead us to this example point
here OF100,

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which was in our but for which 2 was not
true.