In Awodey's book Category Theory, he asks if we have a category by taking "sets as objects and as arrows, those $f\, :\, A \rightarrow B$ such that for all $b \in B$, the subset $f^{-1}(b)\subseteq A$ has at most two elements (rather than one)". He seems to suggest the answer is "yes", but if you take $f$ to be a function from a 3-set to a 2-set and $g$ to be a function from a 2-set to a 1-set, doesn't $g \circ f$ necessarily fail to be an arrow?

How can I handle taking the square root of a negative fraction?

Im pretty sure I should have a real answer, but so far all answers seem wrong, I have cscΘ and need to get the other 5 trigonometric functions, Θ is in quadrant 3 so all but tan and cot are negative

Could you take a look at this?

@RandomVariable: the animation I've just added to the answer may make things a bit clearer. I've also improved my description a little.

@robjohn Seems that Chris has been missing the whole day.

@ABeautifulMind I wonder what went on. Perhaps she was deep into stuff for her book.

@robjohn I think you should not answer questions in chat anymore. You should tell them to post on main to give you more points.

@robjohn My question is fresh for the pointage :D

@ABeautifulMind why? sometimes people need more individual attention to understand something.

@robjohn Don't take my words too seriously. Most of them are a joke.

@ABeautifulMind Statistically, only 2/13 of your last sentence was "a joke".

@Axoren Glad you talk crap like me.

@ABeautifulMind :P I'm known for terrible puns and disappointingly dry humor.

@JasperLoy I'm failing complex analysis

@Axoren I may in a while, but don't hold your breath. I have not done any of that kind of thing in many years.

@robjohn I'll continue breathing then. Thanks.

@user130018 Hi Bart. Don't worry. I'm failing life. Try again, and forgive yourself.

@user130018 I have faith in you.

@JasperLoy thx

Well, this is embarrassing. 228 Project Euler problems solved, I attempt a 10% difficulty problem (new difficulty ratings implemented), can't solve it. I'm actually starting to think their answer is incorrect, even though so many people have successfully solved it...

@Mike I don't know how to solve the problems where they give you a file

My book is having me match a column with another, and says "You may have to rewrite one or both expressions", and I don't really get it? I mean like simplify or full rewrite what just makes sense? Its very confusing

It says -tan(x)cos(x)=sin^2(x)/cos^2(x)

Tell me a topological space which is completely regular but not hausdorff

and that 1+sin^2(x)=csc^2(x)-cot^2(x)+sin^2(x)

But I don't get why

@user use this

the book says any trivial space with more than one point

but i dont get it

well, what's the definition of completely regular

@MikeMiller $T_{3.5}$ and $T_0$.

@DanielFischer That question was not for you, though I appreciate the answer.

@MikeMiller Just to make sure you're not accidentally using the other nomenclature ;)

And I assume sec^2(x)-1=tan x but this is just like a guessing game, and any of the questions can have any answer because I'm just freely rewriting it to a correct form and not doing anything to them? Am I doing this wrong?

I thought nomenclature was a fruit for some reason

for example if I take $X=\{a,b\}$ and define $f(a)=0$ and $f(b)=1$

$f$ is not cointinuos

Why do you say that?

You have to have a topology on X.

@MikeMiller But, that means a space with the trivial topology is only completely regular if it contains at most one point. So the course/book of @user uses the other (wrong, wrong, wrong) nomenclature :(

because taking the trivial topology

$f^{-1}((1/2, 1])=\{1\}$ which is not open

but $(1/2,1]$ is open in $[0,1]$

The term trivial topology is awful. I guess yours means the indiscrete topology.

@Maximilian sec^2(x)-1=tan^2(x). That's an identity.

yes

$\tau=\{X, \emptyset\}$

that's the example that the author gives

of a $T_{3.5}$ but not $T_0$ space

@Mike yes, but its not written like that in the answer column

so I don't get it

or maybe I should use another function

but I don't know which

@user Is that a typo for $T_{3.5}$, or what is $T_{1.5}$?

and It says I may need to rewrite one or both of them, so it seems like a guessing game because I can change the question of 44 and fully rewrite as question 45 or so? There should be some way to know which answer is which

sorry

i meant 3.5

No, you will always have this problem if this is the definition of completely regular.

They have tan x on the side, so I just pick the correct one and slap on the ^2?