# Search results for query: *

1. ### MHB How many miles did she drive

You can see immediately that something is missing. Imagine you can find a solution $D$ for the distance. How would you known if this represents miles, kilometers, or some other unit ?
2. ### MHB Definition of onto function

Point 1 was correct. $f(A)$ is the range, $B$ is the co-domain. $f$ is onto if the range equals the co-domain, i.e., if $f(A) = B$.
3. ### MHB Definition of onto function

Regarding point 4, $f^{-1}(B) = \{x \in A \mid f(x)\in B\}$. Since B is the co-domain, this is true for any function $f:A\to B$.
4. ### MHB Group homomorphism

Hi lemonthree, That is true, but you should also check that $\phi(x^{-1}) = \phi(x)^{-1}$ $\phi(1) = 1$
5. ### MHB Guide to Dealing with Double Subscripts

Hi Peter, You should use braces, like $E_{k_1}$.
6. ### MHB Passengers on the bus

Hi anemone, Is there information available somewhere about the Singapore method ?
7. ### MHB The Union of Two Open Sets is Open

The point is that the argument is valid for every $x\in A_1\cup A_2$. If $C = A_1\cup A_2$, we have proved that, for every $x\in C$, there is an open ball $B(x,r)\subset C$ (where $r>0$ depends on $x$). That is precisely the definition of an open set.
8. ### MHB -gre.al.9 absolute value domain

The distance between $y$ and $-3$ is at most $4$. That means $y$ is between $-3-4=-7$ and $-3+4=1$.
9. ### MHB Find the diameter of one circle

I would say that the sums of the gaps are 24 (above) and 60 (below). As there is one more semicircle above, its diameter is equal to the difference 36.
10. ### MHB Proving Z[x] and Q[x] is not isomorphic

Hi again, In fact, it is even simpler. If $\theta:\mathbb{Z}[x]\to\mathbb{Q}[x]$ is an isomorphism, then $\theta(1) = 1$, because any ring homomorphism must map $1$ to $1$. Now, in $\mathbb{Q}[x]$, we have $1 = \dfrac12+\dfrac12$. If $f(x)=\theta^{-1}(\dfrac12)$, we must have $f(x)+f(x) = 1$...
11. ### MHB Proving Z[x] and Q[x] is not isomorphic

Hi Cbarker1, I don't see what is the point of your function $\phi$: it is not even defined on the whole of $\mathbb{Z}$. In reference to the title of you post, to prove that $\mathbb{Z}[x]$ and $\mathbb{Q}[x]$ are not isomorphic, you could use the fact that $\mathbb{Q}[x]$ is a Euclidean...
12. ### MHB Find the total number of red and blue beads

Hi anemone, Are you sure the problem is correctly stated ? As I read it, you should still end up with $\dfrac25$ blue beads in container A. As the proportion does not change, you must move $3$ red beads for every $2$ blue. However, this can only decrease the proportion of blue beads in...
13. ### MHB Find the last digit of a series

We have: \begin{align*} S_1 &= 1 + \cdots + n = \dfrac{n(n+1)}{2}\\ S_3 &= 1^3 + \cdots + n^3 = \dfrac{n^2(n+1)^2}{4} \end{align*} This shows that $S_3 = S_1^2$. Therefore, if $S_3\equiv1\pmod{10}$, then $S_1\equiv\pm1\pmod{10}$. It is rather obvious that $S_1\equiv S_3\pmod2$. We may write...
14. ### MHB Calculating Truth Tables for Propositions

Hi evinda, That looks correct
15. ### MHB Real Roots of Polynomial Minimization Problem

Since $x=\dfrac{n+1}{2}$ is an axis of symmetry, the point $x=\dfrac{n+1}{2}$ is either a minimum of a maximum, depending on the shape of the quartic. However, the derivative $f'(x) = 4\left((x-1)^3+\cdots+(x-n)^3\right)$ is an increasing function (since it is a sum of increasing functions)...
16. ### MHB Numbers with a quadratic property

As a matter of fact, I was interested in that very question. The question is about finding points with integer coordinates on a hyperbola, and this is a classical problem on representation by quadratic forms. I wrote something about it here. Sorry, it's in French, but ‶the equations speak for...