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comment Is $\Bbb Q[x]/(x^2+x)$ isomorphic to $\Bbb Q[x]/(x^2-x)$?
@mathshungry Because $x\mapsto -x$ takes $x^2+x$ to $x^2-x$. And yes for your interpretation.
comment What should I have learnt as an undergraduate?
I'd say, You should know where to find information you may lack ...
comment Prove that the sum of the squares of two odd integers cannot be the square of an integer.
This is quite fine. For completeness you might want to add that an even square must be the square of an even number / divisible by 4.-- Anyone more experienced might have remembered that odd squares are $\equiv 1\pmod 8$, hence the sum of two such is $\equiv 2\pmod 8$, which cannot be square. Your argument boils down to working $\pmod 4$, which is in fact sufficent
answered Is $\Bbb Q[x]/(x^2+x)$ isomorphic to $\Bbb Q[x]/(x^2-x)$?
revised Why formulate continuity in terms of pre-images instead of image?
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comment is this $f(x)$ continuous at almost every point of an interval $[a, b]$.
Of course, why not?
comment What is the difference between ≎ and ≏?
See also…
answered What is the logic underlying this proof?
comment $a,b,c,d\ne 0$ are roots (of $x$) to the equation $ x^4 + ax^3 + bx^2 + cx + d = 0 $
@πr8 If I'm not mistaken, we arrive at $2b^3a^4 + (b^4 + 2*b)a^3 + 3b^2a^2 + (b^3 - b + 1)a + b=0$ as condition for $a,b$. Playing around it seems that e.g. $b=1$ allows rational solutions for $a$ namely $a=1|-1|\frac12$, then $c=-1|1|2$. Only the last case leads to non-zero $d$. I have not guessed other solutions yet
comment Is $\{ a_{m}(n) | a_{m}(n)$ with all even digits$\}$ finite set for all $m$?
So a simpler(?) preliminary question might be: Is there any $m$ for which it is known whether $X_m$ is finite or infinite?
comment Die that never rolls the same number consecutively
Interestingly, the third roll is a bit more likely to equal the first roll ...
answered Prove that $a . g = g^{-1}.a$ and $g . a = a . g^{-1}$ hold for any group and any action defined.
comment Is $\{ a_{m}(n) | a_{m}(n)$ with all even digits$\}$ finite set for all $m$?
Do you even know any $m$ for sure where $X_m$ is finite?
answered How to differentiate something with u and x?
comment Proof for Urysohn's lemma.
But the intuition part of that lemma is not that hard, I suppose: The idea is to construct a continuous functoin $\to[0,1]$ as a limit of finer and finer staircase functions. The ability needed to refine a staircase function is precisely the condition in the premise of the lemma.
comment Find a **bijection** between two intervals
Once the intervals are corrected, try maps of the form $x\mapsto ax+b$
comment How to Calculate Flat rate using Effective Rate?
If this is about nominal interest vs. effective interest, this is confusing - a factor near 2 is nearly impossible for bearable rates
comment Can we find prime numbers with any sum of digits (except those divisible by three)
The approximately $10^n/n \ln 10$ primes below $10^n$ all have $sd_{10}\le 9n$, so there should be plenty of opportunities to "hit" any allowed value ...
comment Straight Edge - Only Geometric Construction
@Henry I'll have to think about that with a real sketch before me - I just took the validity of the construction in the referenced answer for plausibly granted ...