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  1. F

    Proof about cycle with odd length

    s is odd, so we can write s = 2k + 1, where k is an integer. We note that ##\alpha^n## is a cycle of length s for all integers n. Then ##\alpha = \alpha^{s+1} = \alpha^{2(k+1)} = \alpha^{k+1}\alpha^{k+1}##. so ##\alpha## is the square of ##\alpha^{k+1}##.
  2. F

    Proof about cycle with odd length

    Thanks for the help and correction, I agree that (12345)12345) = (13524) So we can write ##\alpha^na_i = a_j## where ##j = (i + n) \mod s## Therefore ##\alpha a_i = a_{((i + 1) \text{mod s})}## and ##\alpha^{s+1}a_i = a_{((s+1+i) \text{mod s})}## Since (i + 1) = (s + 1 + i) (mod s) we have...
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    Proof about cycle with odd length

    Homework Statement In the following problems let ##\alpha## be a cycle of length ##s##, and say ##\alpha = (a_1a_2 . . . a_s)##. 5) If ##s## is odd, ##\alpha## is the square of some cycle of length s. (Find it. Hint: Show ##\alpha = \alpha^{s+1}##) Homework Equations The Attempt at a...
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    Powers of permutations

    OK i made a mistake reading the question, sorry about that, but think I can use post 3's hint.. We observed that raising a permutation, ##\alpha## of length s, to the ##s^{th}## power, we have ##\alpha^s = e##. Therefore ##\alpha^{-1} = \alpha^{-1}e = \alpha^{-1}\alpha^s = \alpha^{s-1}## []
  5. F

    Powers of permutations

    Homework Statement In the following problems, let ##\alpha## be a cycle of length s, say ##\alpha = (a_1a_2 ... a_s)## 3)Find the inverse of ##\alpha## and show that ##\alpha^{-1} = \alpha^{s-1}## Homework Equations I've observed in the previous problem that there are ##s## distinct powers of...
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    Integer programming model (alternating constraints)

    Thanks for the reply. Since we chose the "loosest" constraint's upper bounds, ##U_1 \ge x_1, U_2 \ge x_2## in constraint 3, that means they are also upper bounds of the first constraint's variables and the third constraint's variables, since an upper bound is just some number that is greater...
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    Integer programming model (alternating constraints)

    Homework Statement Formulate as a mixed integer programming problem but do not solve. Maximize ##x_1 + x_2## subject to ##2x_1 + 3x_2 \le 12## or {##3x_1 + 4x_2 \le 24## and ##-x_1 + x_2 \ge 1##} ##x_1, x_2 \ge 0## Homework Equations The Attempt at a Solution if the first constraint is met...
  8. F

    Knapsack problem constraints help

    This all makes sense, ill keep it in mind on the rest of this chapter. Also i'll probably make a note next to the constraint to clarify that I think we can bring at most one snack. Thank you.
  9. F

    Knapsack problem constraints help

    That would mean changing the objective function to Maximize ##x_4x_1c_1 + x_4x_2c_2 + x_4x_3c_3 + x_4c_4 + x_5c_5 + . . . + x_nc_n##
  10. F

    Knapsack problem constraints help

    Ok so the constraint would be ##x_1 + x_2 + x_3 \le 3x_4## This means if we don't bring the can opener, then ##x_4 = 0##. Therefore ##x_1, x_2, x_3 = 0##. I was hesitant to use this because I thought there would be cases where we brought the unopennable items without the can opener, so their...
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    Knapsack problem constraints help

    I think if say.. I brought 2 identical toothbrushes, then they would just count as 2 separate items with the same weight/volume/value. In total we get ##n## items, but I don't think they have to be unique. The values/weights/volumes aren't specified.
  12. F

    Knapsack problem constraints help

    Homework Statement 8 (a) (The Knapsack Problem) A backpacker's knapsack has a volume of V in.^3 and can hold up to W lb of gear. The backpacker has a choice of ##n## items to carry in it, with the ##i##th item requiring ##a_i## in.^3 of space, weighing ##w_i## lb, and providing ##c_i## units of...
  13. F

    Can a set include negative infinity and be bounded below

    That is my mistake, I see that −∞ is not in (−∞, ∞). What I meant was "If an interval is approaching −∞, can it also be bounded below?". I'm sorry for the carelessness.
  14. F

    Proof about bounds

    Homework Statement Consider the sets below. For each one, decide whether the set is bounded above. If it is, give the supremum in ##\mathbb{R}##. Then decide whether or not the set is bounded below. If it is, give the infimum. Finally, decide whether or not the supremum is a maximum, and...
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    Can a set include negative infinity and be bounded below

    So basically, if a = -|m| - 200 ##\epsilon## A and a < m, it follows that there is no lower bound, because for all elements m ##\epsilon## A, there exists an a ##\epsilon## A, such that a < m. edit: had some questions.. but rereading your post a few times answered them...
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    Can a set include negative infinity and be bounded below

    Homework Statement Prove that {##x \epsilon \mathbb{R} : x^2 \ge 1##} is "not" bounded below. EDIT: I Looked closely and realized there is a "not" that we all had to write in...sorry if you lost some time.. Homework Equations Defintion: We say a nonempty subset ##A## of ##\mathbb{R}## is...
  17. F

    Max/Min proof I can't follow

    I will keep this in mind while going through this chapter, Thank you.
  18. F

    Max/Min proof I can't follow

    Ok, and to show minimum we would do this: Suppose ##h## is a minimum of ##(0,2)##. Then ##0 < h < 2## by definition of minimum. But ##0 < \frac {h}{2} < h < 2##. Thus h is not a minimum, a contradiction. We conclude that ##(0,2)## does not have a minimum. [] note: for the minimum, we could...
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    Max/Min proof I can't follow

    Thank you, now I get it
  20. F

    Max/Min proof I can't follow

    Homework Statement Give an example of a bounded set that has neither a maximum nor a minimum. (The proof below is given by the book). We claim that the set ##(0,2)## is bounded and has neither a maximum nor a minimum. Proof: For each ##x \epsilon (0,2)##, we know that ##0 < x < 2##. Therefore...
  21. F

    Transitivity question

    Thank you for the suggestion. So using this and the next post, my transitivity step would look like.. (Transitivity) Suppose a ~ b and b ~ c. Then ##\frac {a^k} {b^k} = 1## and ## \frac {b^n} {c^n} = 1## for some ##k, n \epsilon \mathbb{Z^{+}}##. Raising the first equation by n and the second...
  22. F

    Transitivity question

    Yea I see your point that the expansion into a + bi is unnecessary.. Here is the updated proof, Proof: (Reflexivity) Suppose ##a \epsilon \mathbb{C}##. Then ##a^k = a^k##, so a ~ a. So ~ is reflexive. (Symmetry) Suppose a ~ b. Then ##a^k = b^k## so ##b^k = a^k##. So b ~ a. So ~ is symmetric...
  23. F

    Transitivity question

    Homework Statement If a, b ##\epsilon \mathbb{C}##, we say that ##a## ~ ##b## if and only if ##a^k = b^k## for some positive integer ##k##. Prove that this is an equivalence relation. Homework Equations The Attempt at a Solution Proof: (Reflexivity): Suppose ##a \epsilon \mathbb{C}##...
  24. F

    Quantifer Proof

    Thanks for the response The z in the antecedent was supposed to be an x. I understand your second point, I made a(a bunch) of mistakes typing what I had. To your third point, this was another typo.. here is the corrected version: original statement to negate: ##\forall x ((x \in \mathbb{Z}...
  25. F

    Quantifer Proof

    Homework Statement Consider the following statement: ##\forall x, ((x \in \mathbb{Z} \wedge \neg(\exists y, (y \in \mathbb{Z} \wedge z = 7y))) \rightarrow (\exists z, (z \in \mathbb{Z} \wedge x = 2z)))## a)Negate this statement. b)Write the original statement in English. c) Which statement is...
  26. F

    Polynomial Proof

    I hadn't thought about that, if we let x-2y = -5 and x-y = -2 then x = 1 and y = 3 which is a solution to the problem. (1-6)(1-3) = (-5)(-2) = 10 if we let x-2y = -10 and x-y = -1, then x = 8 and y = 9 which is a solution. (8-18)(8-9) = (-10)(-1) = 10 I forgot to add my actual answer.. since...
  27. F

    Polynomial Proof

    then x-y is larger.. so from my above post, I would automatically know x-y =5 or x-y = 10 Ok so if x-y = 10 then x-2y = 1. Solving these equations gives x = 19 and y = 9. We confirm this by (19-9)(19-2(9)) = 10*1 = 10. So this is one solution. If we let x-y = 5 then x-2y = 5. Solving these...
  28. F

    Polynomial Proof

    Thanks for the response, I think you did the proving when made x=8 and y=3. I made a mistake.. i'm not sure what the x intercepts represented when I graphed it.. I think since I couldn't just see it, I could have set ##x-2y = 1,2,5,## or ##10## and then set ##x-y = 1,2,5## or ##10## depending...
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