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Mar
13
comment How to call a mathematical space “$(\mathcal S, f)$” consisting of set $\mathcal S$ and function $f : \mathcal{S \times S} \rightarrow \mathbb R$?
@user12262 you are, of course, free to study whatever you like. If you accumulate enough results that merit the study worthy, great. Then it would be a good time to think of a suitable name for your structure. It is rather useless to spend some much time initially, before you have any justification for your study, in search of a name. Go ahead and investigate. Personally, I don't see the point in the study of a set together with a function as you describe. Perhaps you have some convincing reasons to explore these things. If so, what are they? Merely "others studied other things" is no reason.
Mar
13
comment Is it possible to prove uniqueness without using proof by contradiction?
Yes, the function $f$. You appealed to its injectivity to prove the uniqueness of the solution. But the two claims are trivially equivalent, so you have to prove one of them. You might as well have said "Suppose we want to prove that the solution to $2x+1=0$ is unique. Well, it's solution is unique, so there you go".
Mar
13
comment Is it possible to prove uniqueness without using proof by contradiction?
and how do you prove that that function is injective?
Mar
8
comment The probability that the equation will have real roots is
Please avoid dropping your questions on us. What have you tried? where did you get stuck?
Mar
8
comment Showing that if $u$ and $v$ are vectors in 2d-space or 3d-space, $∥u+v∥≤∥u∥+∥v∥$
Draw the vectors $u$ and $v$ as arbitrary vectors from the same origin, forming the sides of a triangle. Realise the length of the third side in the triangle is $\| u+v\|$. What do you know about the length of a side in a triangle compared to the sum of the lengths of the other two sides?
Mar
6
comment Some Trouble Understanding set theory
With time you will develop intuition. For now, plugging in sets and seeing what happens is a good way to build intuition, and sometimes solve the problem. Working with Venn diagrams also assists intuition. You can also try to formally prove a statement, and either succeed (yey!) or fail, and see where the proof fails, hinting at a counter example.
Mar
6
awarded  Guru
Mar
4
comment Intuition on proof of Cauchy Schwarz inequality
@Chan-HoSuh my comment was about a conceptual problem. There is no problem in the proof (in fact its my favourite proof). All I said was that before you interpret orthogonality in the geometric sense as "inner product is $0$" you need angles. And before you have CS, you don't have angles. Of course, algebraically it all works out, and the intuition is vindicated after the fact. And that's fine. That's all I said.
Mar
3
asked closed union of closed sets
Mar
3
comment In the Cantor diagonal argument, how does one show that the diagonal actually intersects all the rows in an infinite set?
you are confusing yourself with completely irrelevant details, but even with that confusion, if your intuition is, for whatever reason, that you get a table of dimensions one countable by two countables, that's fine too, since it is a square matrix (since countable + countable = countable).
Mar
3
answered In an extension field, is there any difference between the original field and its isomorphic copy in the extension field?
Mar
1
comment How can there be an inner product space when inner product yields a scalar?
It is entirely unclear what you are asking. Please significantly improve your question.
Mar
1
comment Sets cardinality in linear algebra
you are drowning yourself in irrelevant details and so I threw you a lifeline. The fact that $V=\mathbb R\langle T\rangle$ and that $W=\mathbb R\langle S\rangle$ is quite irrelevant. The only thing that matters here are the dimensions.
Mar
1
answered Sets cardinality in linear algebra
Feb
27
comment Quotient topology and the need for continuity
continuity is the whole point of topology. Topologies exist so that we can define what it means for functions to be continuous. From the point of view of topology, nothing is more natural than to ask "what is the largest topology on the quotient so that the quotient map is continuous?". What happens if you do not demand continuity? well, then you are not doing topology but something else, and then the answer depends on what is that something else you are doing.
Feb
27
revised monoid action on categorical limits
added 3 characters in body
Feb
27
asked monoid action on categorical limits
Feb
24
comment Theorems Implying their Own Generalization
how do you justify that $1+2+\cdots + n$ is a polynomial in $n$?
Feb
24
revised Can we get categories like $\mathbf{TopGrp}$ as some kind of a pullback?
deleted 1 character in body
Feb
23
answered Theorems Implying their Own Generalization