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  1. K

    Prove Set of all onto mappings from A->A is closed

    Still not very sure on how to start. BTW, is there anything rescuable from my first approach?
  2. K

    Prove Set of all onto mappings from A->A is closed

    Thanks. We need to show that ##fog## is onto or in other words fog = z = f(g(x)) =A? f: A-> A : f(a) = A g: A ->A: g(a) = A Is this correct?
  3. K

    Prove Set of all onto mappings from A->A is closed

    Homework Statement Prove that set of all onto mappings of A->A is closed under composition of mappings: Homework Equations Definition of onto and closure on sets. The Attempt at a Solution Say, ##f## and ##g## are onto mappings from A to A. Now, say I have a set S(A) = {all onto mappings of A...
  4. K

    Show that is not onto (##\frac{x}{x^2+1}##)

    Once, I have ##y(x^2+1) = x## what else can I do? ## y x^2 + y = x## divide all over x^2 wouldn't work.
  5. K

    Show that is not onto (##\frac{x}{x^2+1}##)

    The question says that A= R-{0} and B =R. Then, that f:A ->B and I need to show whether they 1-1 and whether they are onto. Prove. Thanks for the hint.
  6. K

    Show that is not onto (##\frac{x}{x^2+1}##)

    Homework Statement I need to show that $$\frac{x}{x^2+1}$$ is either onto or not. My domain is $$R-{0}$$ and range is $$R$$ Homework Equations I have learn to do this to show that a function is surjective y = $$\frac{x}{x^2+1}$$ and solve for x, but I am not sure how to proceed here. The...
  7. K

    Bi-variate Non-Homogeneous Polynomial Conceptual Question

    Homework Statement I have found the roots of my polynomial: ## (2x+3y)^{2}-1 =0 ## Roots are x=3n+2 & y=-2n-1, where n belongs to all Z. What does it mean that the solution has arbitrary large coordinates? The Attempt at a Solution I think I know the basic concept of root. It could be...
  8. K

    Compute how many n-digit numbers

    So, here is the solution. ## \dfrac{ |n| !}{ |n_{1}|! |n_{2}|!... |n_{k}|!} \text{, where n is the size of the vector and the values in denominator are types of symbols in n that repeat.}## ##\text{For instance, if I have a vector called v={e, i, g, e, n, v, a, l, u, e}, there are say n1 type...
  9. K

    Compute how many n-digit numbers

    Please, check the solution in attachment. Apparently, it is incorrect. Can someone verify? I think that I am not taking into account cases such as {m, o ,m } or {1,0,1}, where there could be repetitions. The solution should be in the form: Order matters { ... , <at least digits from 0 to...
  10. K

    Compute how many n-digit numbers

    It turns out that order does matter after all. One 'gotta' love language. An example helped to elucidate. Example: {0,1,2,3,...,8,9, ...<whatever>,...} is a vector of size n and 1 solution that satisfies the constraints. {3,4,7,...,9,2,1,...<whatever>,... } is another vector of size n and...
  11. K

    Compute how many n-digit numbers

    Let me get back to you in a few days, I believe I can obtain more specific information about the problem. Thank you for your help so far.
  12. K

    Compute how many n-digit numbers

    The question is for a Theoretical Computer Science class. You might be right, but he said that it does not matter. My professor was very vague in posing the question. I asked it twice. He gave some examples: ## {0, 0, 0,0, ...,0_{n} }={0} ## ## {0, 0, 0, ..., 0, 1_{n}}={0,1}## Yes, it is...
  13. K

    Compute how many n-digit numbers

    Homework Statement Compute how many n-digit numbers can be made from the digits of at least one of {0,1,2,3,4,5,6,7,8,9 } Assume, repetition or order do not matter. Homework Equations ## a_{1}, a_{2}, ..., a_{n} ## The Attempt at a Solution 10 choices for the 1st sub-index, 10 choices for...
  14. K

    -arctan(x/y) = arctan(y/x) ?

    Homework Statement -arctan(x/y) = arctan(y/x) ? Are they equivalent? I can't find anything online and I have seen that my solution to some problem involves -arctan(x/y) and it agrees with Wolfram Alpha. On the other hand, my professor's solution shows the arctan(y/x) and this is why I am...
  15. K

    Separate Variables Differential Eq. of Cubic Power

    Thank you, tiny-tim. I usually use latex for big equations, but I thought it wouldn't be a big deal. I was thinking that I could solve it like your wikipedia link... this will be interesting. Thanks.
  16. K

    Separate Variables Differential Eq. of Cubic Power

    Yes, I am aware that it is kinda difficult to solve for 'y' and that's why I wanted to try it out. It involves imaginary numbers and many roots. If someone can point me in the right direction, that would great.
  17. K

    Separate Variables Differential Eq. of Cubic Power

    Homework Statement When possible express the general solution in explicit form. Solve dy/dx =x^2 /(1+y^2) Homework Equations This is a first order non-linear ordinary differential equation. The Attempt at a Solution dy(1+y^2) = x^2 dx Integration both sides returns: y+ (y^3 )/3=...
  18. K

    Elliptic Area Using Integral

    I know that the axis are rotate with respect of the x-y coordinates. I thought that the equation was still valid in these cases. Yes. I see the xy term and I showed it in the first post. I am trying to compare it with 1, since it fits the equation. However, this equation would be wrong. I...
  19. K

    Elliptic Area Using Integral

    Homework Statement (x+y)^2 + (y-2)^2 =4 2. Homework Equations y^2 = (2-x)y - (x^2)/2 Equation of an ellipse: ##\left( \dfrac {x-h} {a}\right) ^{2}+\left( \dfrac {y-g} {b}\right) ^{2}=1## From this, we know that (h,g) is the center of the ellipse. and the radius along the x and y...
  20. K

    Quadratic Residues Modulo p

    I didn't say that b1-b2=1, I meant that b1=b2=1. So, supposedly since p|(b1-b2); b1-b2= kp, where k is an integer. Are they saying that it's impossible p | (b1-b2) , unless b1=b2 or in other words that p|0 or 0= pk, where k is any integer and in this case k=0? Or are they saying that...
  21. K

    Quadratic Residues Modulo p

    Homework Statement Let p be an odd prime. A number ##a\in \mathbb{Z} _{p}^{\ast }## is a quadratic residue if the equation ##x^{2}=a\left( \mod p\right)## has a solution for the unknown x. a. Show that there are exactly (p-1)/2 = quadratic residues, modulo p. The Attempt at a Solution...
  22. K

    Solutions to Congruence Modulo 50

    According to Wikipedia, Diophantine equations are written as follows: ax + by = c The Diphantine equation that you are really writing is this 35x-50n=10? I understand everything, until you change the equation 1=7 -2(10-7)= 3(7)-2(10)=1. I understand that 21-20=1, but why changing from 7-...
  23. K

    Solutions to Congruence Modulo 50

    I have read somewhere that division is not defined in modular arithmetic. Can someone tell me how this affect my solution? @kru: This is puzzling, since I found those other solutions at a .edu site.
  24. K

    Solutions to Congruence Modulo 50

    Homework Statement Find all solutions to the equation ##35x\equiv 10mod50## The Attempt at a Solution gcd( 35,50)= 5 So, there is a solution to this, since 5| 10. Also, there is a theorem that guarantees the existence of exactly 5 solutions. Now, dividing ##35x\equiv 10mod50## over...
  25. K

    Find Intervals, where Function is Convex or Concave and Inflection Points

    Homework Statement y= (x^2 -7) e^x The Attempt at a Solution I'm trying to find inflection points by setting the second derivative=0 I found that the derivative is: ##2xe^{x}+x^{2}e^{x}-7e^{x}=0## ##e^{x}[2x+x^{2}-7]=0## Then, the 2nd derivative: ##e^{x}[(x-1)(x+5)]=0##, then the...
  26. K

    Limits and Vertical Asymptotes

    Thank you very much to all. Now, it remains clear that the same principle of limit is there. Thank you, again!
  27. K

    Limits and Vertical Asymptotes

    What do you mean by, "the denominator approaches zero, but remains constant"? How do you know that it remains constant? How so?
  28. K

    Limits and Vertical Asymptotes

    The denominator is never negative. It approaches 0 from the right hand side. Is this the same as saying: (any constant)/ (a number infinitely small that never really reaches zero) ? Then, it follows by the same principle as 1/ (x^2) as x->0 grows without bound?
  29. K

    Limits and Vertical Asymptotes

    I see that the numerator is approaching 2.333... and the denominator approaches 0. I know that, when lim x-> 0 1/(x^2) it's very clear that the function grows without bounds, but in this occasion I just can't see how it grows.
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