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  1. Implication in Propositional and Predicate Logic

    Below are listed several pairs of sentences, both in propositional logic and in predicate logic. In each pair, one of the sentences implies the other. Decide which of the two logically implies the...
  2. Writing matrix A = [[0 -2][7 -6]] as the product of elementary matrices

    \mbox{Let }\, A\, =\, \left[\begin{array}{rr}0&-2\\7&-6\end{array}\right]

    (i) Write A as a product of 4 elementary matrices.

    (ii) Write A^-1 as a product of 4 elementary matrices.

    I have 94%...
  3. Finding elementary matrices to complete a set of matrix equations

    9094
    I have 89% of the values correct but I'm not sure where I'm going wrong exactly.

    For the first ones, the first row is being multiplied by 2 and then multiplied by 1/2, so I multiplied the...
  4. Chinese Remainder Thm/Simult. congruences: X ≡ 2 mod 3, X ≡ 2 mod 4, X ≡ 1 mod 5

    "Find an integer X such that


    X ≡ 2 mod 3,


    X ≡ 2 mod 4, and

    X ≡ 1 mod 5."
  5. Finding limit of Riemann Sum (1/n) sum[k=1,n] cos(4k pi / 3n)

    Problem 5: Express the following expression:

    . . . . .\displaystyle \dfrac{1}{n}\, \sum_{k=1}^n\, \cos\left(\dfrac{4k\pi}{3n}\right)

    ...as the Riemann sum for an integral of the form:

    . . ....
  6. Calculating the integral of ((-x^2)/3)-2, from 0 to 2, using the limit definition

    "f(x) = ((-x^2)/3)-2

    We are supposed to calculate the integral from 0 to 2 of f(x) using the limit definition of an integral.

    a) Calculate Rn for f(x), which is the Riemann sum where the sample...
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    Basis for the Null Space of a Matrix

    "Let A = \begin{pmatrix}-1&2&-13&-6\\ 2&1&1&-3\\ -3&0&-9&0\\ 7&2&11&-6\end{pmatrix}


    Determine a basis for the null space of A."


    So I reduced it and got the following matrix:

    ...
  8. Yes, exactly! I think this was supposed to be one...

    Yes, exactly! I think this was supposed to be one of the harder ones.
  9. OK I understand. Thanks for clarifying!

    OK I understand. Thanks for clarifying!
  10. Wow, that's interesting! I never knew that. This...

    Wow, that's interesting! I never knew that. This would have really simplified everything. Thanks for the advice! I will keep it in mind from now on.
  11. Yes, the "strange numbers" I was referring to...

    Yes, the "strange numbers" I was referring to were the very large ones requiring scientific notation. This put me off a little bit. But I just need to be more careful next time, that's all.
    ...
  12. I see. Yes the textbook's method is fine...

    I see.

    Yes the textbook's method is fine (matches the answer expected in software). I just overlooked it because this particular function is very weird and involves some strange numbers so I got...
  13. The software you are describing sounds quite...

    The software you are describing sounds quite advanced. Ours seems to be quite simple in comparison. But we do have a textbook through which we can read up on any course material. So, honestly, its...
  14. Thank you! Your answer is correct! I guess I...

    Thank you! Your answer is correct! I guess I misinterpreted the concavity at -4 and 4.

    Yes it was a piece of software on which I was doing this. It tells you when your answer is right or wrong but...
  15. Where is function f(x) = (x^2 - 16)^6 concave up?

    The function is ((x^2)-16)^6.

    So I did all the relevant calculations and all my others answers about inflexion points, critical points, extreme points, concave down, increasing and decreasing...
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    Finding the matrix T(x)

    "Let T be a linear transformation on the plane with

    T \begin{pmatrix}2\\ 3\end{pmatrix} = \begin{pmatrix}-3\\ 3\end{pmatrix} and T \begin{pmatrix}3\\ 4\end{pmatrix} = \begin{pmatrix}5\\...
  17. Rate of change of distance: plane w/ constant speed 19km/min passes 10km over radar,

    "A plane flying with a constant speed of 19km/min passes over a ground radar station at an altitude of 10km and climbs at an angle of 20 degrees. At what rate is the distance from the plane to the...
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    Rate of change - Implicit differentiation

    "A price p (in dollars) and demand x for a product are related by


    (2x^2)-2xp+50p^2 = 20600.


    If the price is increasing at a rate of 2 dollars per month when the price is 20 dollars, find...
  19. OK, that makes sense now. Thank you for your help!

    OK, that makes sense now. Thank you for your help!
  20. Yes, perhaps I made an error in that step. But I...

    Yes, perhaps I made an error in that step. But I still keep getting the same eigenvector.

    This is what I did for the eigenvalue of 6:

    \begin{pmatrix}2&2\\ \:-4&8\end{pmatrix}...
  21. Thank you very much for your helpful reply! I...

    Thank you very much for your helpful reply!

    I did what you said and got the matrix half correct. Two of my entries still seem to be incorrect. Here is what I did:

    P =...
  22. I see. So would that mean the eigenvalues of the...

    I see. So would that mean the eigenvalues of the new M^n matrix will be 4^n and 6^n? If so, do I apply this formula to the entries of the initial matrix?

    Thank you!
  23. Thank you for your reply! But I am sorry I do not...

    Thank you for your reply! But I am sorry I do not understand what you are referring to when you say common factors. There does not seem to be anything I have written above that seems to require...
  24. General formula for multiplying a matrix by a power

    "Let M = \begin{pmatrix}2&2\\ -4&8\end{pmatrix}


    Find formulas for the entries of M^n, where n is a positive integer."

    So I think this problem involves multiplying a matrix by itself...
  25. Yes, I have managed to solve it now. I was still...

    Yes, I have managed to solve it now. I was still a little confused at that point so I thought it would be better to ask but I manged to figure it out. Yes, the quadratic formula was useful. Thank...
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