so i have a pretty good story or maybe you can say a pretty bad story
with this old SAT math question here and because i really want to make this video
i actually bought the book so i can find the same question again this question is
actually from one of the previous actual sat subject test math level 2. and in my opinion
this is a real analysis question and the sat is meant to be for high school students when they
are applying to college and i first encountered this question when i was volunteering teaching
an SAT prep class at UC Berkeley via student group called the PTPS people's test preparation
service and i was trying to explain this question to the students i couldn't do it let's have a look
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so do you think that this question is confusing especially if you have never taken like a proof
type of math class before i think so and you know what's worse the answer for this question is none
not only the language is just so mathematical but also the answer is none none of these
okay so let's go over how to answer this question first off what is the question
asking which set has an element that is less than any other element in that set
this is how math people will write the question but the truth is we can actually read this in a
simpler language the question is just asking which set has a smallest element that's it
because we want an element to be smaller than any other element so that element has
to be the smallest that's it and of course we have these options and just for practice
i wanted to show you guys how we can use the step builder notation to write this set for the first
one we want a set of past rational numbers so here's the set builder notation and let's use r
and first we want the r to be positive so we can say r is greater than zero and then we can put
on vertical bar meaning such that and then we are going to put down the property of what r shall be
r has to be a positive rational number we have the positive right here already and then for r to be a
rational number it means it has to be an integer over an integer so we can write this as a over b
but we have a fraction so that means we can have 0 on the bottom so we'll say b cannot be equal to
0 and then we also have to say a and b they are elements right that's what this notation is for
the set of integers so that's what the z is for and close that in fact this set has a special
notation the set of all the rational numbers is just q denoted by this and if you want just the
positive rational numbers you can put on q with a plus like so all right now for number two we want
the set of past the rational numbers so we can say r and we can say that first we want r to belong to
q plus so that means r is a rational number and that's also going to be positive
such that we want r square to be greater than or equal to 2 so we'll put that down right here
so that would be the stability notation for that and then the third one we want
to set again past the rational number so r belongs to q plus such that we want r square
to be greater than four so let's go back to here do we have a smallest positive rational number
the answer is no why because the answer is not of course i'm just kidding here's the deal all
right let's think about maybe we have let's say 1 over 10 it's a pretty small rational number but
is this the smallest no because if you say this is similar i can prove it wrong i can say 1 over
100 this is going to be a smaller rational number than 1 over 10. well if you say this
just put down 1 over 1000 then you see that that's not the smallest this is smaller you see
you can keep on going forever unfortunately you are just not going to find
a smallest positive rational number if he says noun negative right if he says now negative
if this is r is greater than or equal to zero in fact this set does have a smallest element
and the smallest element is zero because right here you see r is greater than or
equal to zero so r can be equal to zero so zero belongs to this set and 0 will be the smallest
now negative rational number okay now let's take a look at number two and per our discussion
earlier because we have the equal sign here so we should have a smallest element right no
because imagine you want to solve this inequality take the squares on both sides then you are saying
r has to be greater than or equal to square root of 2 just a positive version because r is positive
okay again we have the equal sign so square root of 2 should be the smallest element for this
but the answer is no because square root of 2 does not belong to
the set of rational numbers square root 2 is not rational i have a proof of that a long
time ago go ahead and check that you will lie but yeah that's why number two it's not
yeah okay number three with the set of positive rational numbers so all that stuff and r square
is greater than or equal to greater than four yeah okay let's solve for this inequality just
take the positive root so it looks like we have r is greater than square root of four is just two
but you see this time we do not have the equal sign we do not have a smallest rational number
that's bigger than 2. so again this right here does not have a smallest element either
if number 3 was saying this right here would have a smallest element because two belong to
this set and then two in fact it will be the smallest so that's why it's none of this
so what do you guys think yeah i was
extremely confused when i was reading the equation because i was just beginning i was just begin
to learn about proofs and i was not great in teaching because that was like the first ever
time that i was student in front of students and i was just a volunteer teacher yeah so
good memory i would say but also bad memory but anyway hopefully you guys enjoyed this that's it