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The Math Help Thread
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falconxcrunner09
Member
Quick question, currently taking Abstract Algebra and need some help with notation.
"For each element a ∈ G, there exists an inverse element in G, denoted by a^−1, such that
a ◦ a^−1 = a^−1 ◦ a = e."
What does the e represent in this context? Is it the identity element which is essentially 1?
"For each element a ∈ G, there exists an inverse element in G, denoted by a^−1, such that
a ◦ a^−1 = a^−1 ◦ a = e."
What does the e represent in this context? Is it the identity element which is essentially 1?
falconxcrunner09
Member
Cool thanks! : )
Allonym
There should be more tampons in gaming
Can anyone help with this?
Question is; A rancher with 900 ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. The problem wants you to solve for area in terms of W and for some reason I keep getting it incorrect and I'm at a loss as to why.
-Okay, so my thought process is this; A(rea) = L * W and P(erimeter) = 2L + 2W but in this situation, 900 = 5L + 2W.
(900 - 2W)/5 = L -------------> ok, so now you would substitute this into the Area formula and get A= (180 - 2/5 W)(W) or 180W -2/5W^2. I thought that this was correct but for some reason it keeps saying it's wrong. Any advice?
Question is; A rancher with 900 ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. The problem wants you to solve for area in terms of W and for some reason I keep getting it incorrect and I'm at a loss as to why.
-Okay, so my thought process is this; A(rea) = L * W and P(erimeter) = 2L + 2W but in this situation, 900 = 5L + 2W.
(900 - 2W)/5 = L -------------> ok, so now you would substitute this into the Area formula and get A= (180 - 2/5 W)(W) or 180W -2/5W^2. I thought that this was correct but for some reason it keeps saying it's wrong. Any advice?
FortuneFaded
Member
Looks right. Are you sure you are inputting correctly i.e. making sure it's (2/5)*(w^2) rather than 2/(5*(w^2))? Could also try 900 = 2L + 5W.
Allonym
There should be more tampons in gaming
-5/2(w-180)w is listed as the correct answer. It makes little sense to me. I have an idea of why the answer looks like that but when I try to go through why it looks like that, it makes less and less sense to me.
Continuing from 900 = 2L + 5W that FortuneFaded suggested above, solve for L:
L = (900 - 5W) / 2
Hence, the area, in terms of W, becomes:
A = L * W = (900 - 5W) / 2 * W
The expression on the right is the same as -5/2 * (W - 180) * W.
NamikazeBurst
Member
Does anyone have any recommendations for learning Introductory Statistics? I have to take this course for my first year of Software Engineering (2nd year of Engineering). Going by online reviews and how he was in my first lecture, it does not seem like my professor will be of much help. The textbook for the course is "Weighing the Odds: A Course in Probability and Statistics" by David Williams. Going by the reviews for the book and reading some of the first sections, it seems to be complicated and not the best book for an Introductory Statistics course. I believe Calculus is also used in this Statistics course.
I hope my knowledge from my Data and Probability course that I took in grade 12 is helpful.
I hope my knowledge from my Data and Probability course that I took in grade 12 is helpful.
Really simple question.
Trying to find limit of X ----> 1^(+) of the function square root (2x+1) - square root(3)/x-1
I am looking at the solution and unable to grasp going from (2x+1)-3/(x-1)(square root(2x+1)+square root (3))
to 2/(square root(2x+1)+square root (3)
How exactly are we getting to the latter? I'm sure I'm forgetting something basic here.
Trying to find limit of X ----> 1^(+) of the function square root (2x+1) - square root(3)/x-1
I am looking at the solution and unable to grasp going from (2x+1)-3/(x-1)(square root(2x+1)+square root (3))
to 2/(square root(2x+1)+square root (3)
How exactly are we getting to the latter? I'm sure I'm forgetting something basic here.
((2x+1)-3)/(x-1) = (2x-2)/(x-1) = 2?
Not in general, no. For example, the set of all functions from R to R, and the set R itself are both uncountable. But the first is larger (in the usual sense that there is no injection from the first set to the second).
However, if you are working within some smaller universal set, you may be okay. For example, if you are only looking at subsets of the real numbers then every uncountable set has the same cardinality.
stanley1993
Neo Member
Hi Gaf,
Can anyone tell me how I can find the answers to the three questions? Also, suggestions as to what I should study would also be helpful. (meaning, topics I can youtube and learn by myself to be able to do these questions by myself)
Can anyone tell me how I can find the answers to the three questions? Also, suggestions as to what I should study would also be helpful. (meaning, topics I can youtube and learn by myself to be able to do these questions by myself)
Although not a direct answer to your questions, this guy's videos got me an A in both Calc I and Calc II(his calc III videos weren't out when i took it): https://www.youtube.com/playlist?list=PLF797E961509B4EB5Hi Gaf,
Can anyone tell me how I can find the answers to the three questions? Also, suggestions as to what I should study would also be helpful. (meaning, topics I can youtube and learn by myself to be able to do these questions by myself)
stanley1993
Neo Member
Ok, great. I'll check it out. Thanks!
I confirmed with my professor that indeed they are not inherently equivalent.
So I'm glad I didn't waste my time to prove they were both uncountable and just skipped it.
ugh, am I understanding this correctly?
find the number of different strings of length 8 with three distinct letters and five underscores
k so
if everything in the string was distinct, there'd be 8! permutations. there are 5! arrangements with distinct underscores for each arrangement where the underscores are not distinct. applying the same for the letters, I'd get 3! putting all this together gets me 8! / (5!3!)
there are 26 letters in the alphabet and I must pick three distinct letters. P(26, 3) = 26!/23!
using the rule of product, my final answer would be (26!/23!) * (8! / (5!3!)). there are that many ways to arrange a string of length 8 using five underscores and three distinct letters
find the number of different strings of length 8 with three distinct letters and five underscores
k so
if everything in the string was distinct, there'd be 8! permutations. there are 5! arrangements with distinct underscores for each arrangement where the underscores are not distinct. applying the same for the letters, I'd get 3! putting all this together gets me 8! / (5!3!)
there are 26 letters in the alphabet and I must pick three distinct letters. P(26, 3) = 26!/23!
using the rule of product, my final answer would be (26!/23!) * (8! / (5!3!)). there are that many ways to arrange a string of length 8 using five underscores and three distinct letters
stanley1993
Neo Member
Do you know what topics I should study for integral calculus? My calculus coursework is a bit abnormal. My first calc. class was applied calc. Now I'm doing integral calc. I'm pretty sure my applied calc. class covered everything but trig. functions. Whats stumbling me now is finding the limits and integrals of trig functions. I took my last calc. class around two years ago, so I'm not used to any calculus terminology.
There's nothing different about limits of trigonometric functions. After all, trig functions are just functions.
It's just that limits of certain functions can be interesting to study (e.g. as x approaches 0, sin(x) / x approaches 1, instead of being undefined).
The three problems that you posted previously are based on these. You may want to brush up on L'Hopital's Rule.
Integration of polynomials uses formulas for finite sums, so you will study sequences and series.
You will then use limits to extend the sum of finite terms to infinite terms (Riemann integration).
This is how you can arrive at integration rules for polynomials.
stanley1993
Neo Member
Alright. Thank you! I'll check all of this out.
AntraxSuicide
Banned
What do you mean by Intro Stats? I teach a Stats for Non-Math course, which sounds easier than what you're doing, but you're in engineering. The engineering intro stats class at my university would cover calculus in statistics, so you may just have to get comfortable with it.
Hi Gaf,
Can anyone tell me how I can find the answers to the three questions? Also, suggestions as to what I should study would also be helpful. (meaning, topics I can youtube and learn by myself to be able to do these questions by myself)
All of these use the limit mentioned above (as x approaches zero, sin(x)/x approaches 1). This cancels the denominator, which lets you substitute.
NamikazeBurst
Member
I don't really know how to answer this. For some reason there isn't a normal course outline for this course. There is some information posted on the website for the course, but there is nothing specific. It doesn't say the topics that will be covered in the course, just general stuff like "state fundamental concepts of applied probability and statistics" as a course objective. So far we have covered the sample space, events of the sample space, basic probability, and union and intersection. I learned these topics in high school, so I haven't been confused on anything yet, however, the next topic we are covering is random variables, which I don't remember if I have been taught them before.
I don't have a problem with Calculus. I can do integration and differentiation. I don't know how complex the Calculus will be when/if we do finally use it. I just included that in my description of the course because I know there are Statistics courses that don't use Calculus.
I was helping a student at my job with a Discrete Math problem today. The question involved proving a statement with mathematic induction. Now, the proof is much easier to do directly, but the requirement was to do so with mathematic induction. We had a debate if it is technically mathematic induction if you prove a base case, make an inductive hypothesis and assume it, and then just prove it without actually using the inductive hypothesis, is it technically mathematic induction? It technically follows mathematic induction, but it's really just wrapping a direct proof inside an inductive proof. Here is the problem
So this is what we did.
Base Case:
n = 1. A = {1, 2}. 1 divides 2
Inductive Hypothesis:
Assume |A| = k+1, k > 0. Assume there exists two elements of A, a and b, such that a divides b.
Inductive Step:
*This is where we don't do what is expected*
Pigeon Hole Proof:
|B| = (k + 1) + 1. All elements of B are integers that are between 1 and 2(k+1), inclusively.
It can be proven fairly trivially with the Pigeon Hole Principle that there exists two elements, a and b, such that a divides b. This proof does not use the inductive hypothesis.
So technically, we proved P(1) is true, and that P(k) -> P(k+1), k > 0. Which is technically an inductive proof.
So is this technically a mathematic induction proof that isn't following the spirit of it, or is it not a mathematic induction proof at all?
So this is what we did.
Base Case:
n = 1. A = {1, 2}. 1 divides 2
Inductive Hypothesis:
Assume |A| = k+1, k > 0. Assume there exists two elements of A, a and b, such that a divides b.
Inductive Step:
*This is where we don't do what is expected*
Pigeon Hole Proof:
|B| = (k + 1) + 1. All elements of B are integers that are between 1 and 2(k+1), inclusively.
It can be proven fairly trivially with the Pigeon Hole Principle that there exists two elements, a and b, such that a divides b. This proof does not use the inductive hypothesis.
So technically, we proved P(1) is true, and that P(k) -> P(k+1), k > 0. Which is technically an inductive proof.
So is this technically a mathematic induction proof that isn't following the spirit of it, or is it not a mathematic induction proof at all?
No, I would say that's not an inductive proof. You should also explain how the Pigeonhole Principle can be applied, rather than saying the proof is trivial.
To be precise, let me formulate the statement as follows:
P
: Let n be a positive integer. For any set A with (n + 1) numbers between 1 and 2n, there exist two elements a, b from the set, such that a divides b.You already showed the base case, so let me jump to the proof P
=> P(n + 1):Suppose P
is true. We want to show that P(n + 1) is also true, so consider a set A with (n + 2) numbers between 1 and (2n + 2).Now, there are 3 possibilities:
1. Set A contains (n + 2) numbers between 1 and 2n.
2. Set A contains (n + 1) numbers between 1 and 2n, and 1 number that is either (2n + 1) or (2n + 2).
3. Set A contains n numbers between 1 and 2n, and both (2n + 1) and (2n + 2).
In cases 1 and 2, we can use the inductive hypothesis P
to conclude that P(n + 1) is true.You just have to prove case 3---maybe by Pigeonhole Principle, I didn't try.
No, I would say that's not an inductive proof. You should also explain how the Pigeonhole Principle can be applied, rather than saying the proof is trivial.
To be precise, let me formulate the statement as follows:
P
You already showed the base case, so let me jump to the proof P
Suppose P
Now, there are 3 possibilities:
1. Set A contains (n + 2) numbers between 1 and 2n.
2. Set A contains (n + 1) numbers between 1 and 2n, and 1 number that is either (2n + 1) or (2n + 2).
3. Set A contains n numbers between 1 and 2n, and both (2n + 1) and (2n + 2).
In cases 1 and 2, we can use the inductive hypothesis P
You just have to prove case 3---maybe by Pigeonhole Principle, I didn't try.
Why is it okay to only use the inductive hypothesis for the first two cases but not the third?
themagicalkitsune
Member
Probably a long shot, but just in case anyone is familiar with this...
Regarding hom and tensor functors I'm trying to compute the tensor product of a vector space V with itself over Hom(V,V) (the set of linear transformations from V to V, which has the structure of a ring). If V = F the base field, it's easy to see that the product is just F. What if it's n-dimensional, say V isomorphic to F^n? I'm trying to find basis elements for this product and I supsect they're of the form V x V, but I don't think I can just say the product is F^{2n} since obviously that contradicts what I found for n =1..
Regarding hom and tensor functors I'm trying to compute the tensor product of a vector space V with itself over Hom(V,V) (the set of linear transformations from V to V, which has the structure of a ring). If V = F the base field, it's easy to see that the product is just F. What if it's n-dimensional, say V isomorphic to F^n? I'm trying to find basis elements for this product and I supsect they're of the form V x V, but I don't think I can just say the product is F^{2n} since obviously that contradicts what I found for n =1..
The inductive hypothesis P
is true if we have (n + 1) numbers between 1 and 2n.In case 3, there are only n numbers between 1 and 2n, so we couldn't use the hypothesis to prove P(n + 1). We would have to prove case 3 by different means.
I didn't study functors, but wouldn't the product be more like F^{n^2}?
The way you described hom functors as linear maps from V to V reminded me of matrices, and n x n matrices can be thought of as vectors with n^2 entries. You could then use the idea of how basis for matrices are derived/defined to come up with basis for your tensor product.
And I guess what I'm saying is the proof for case 3 can also be used for case 1 and case 2. I guess I feel like this is an odd problem for trying to teach mathematic induction. Here's a similar problem.The inductive hypothesis P
In case 3, there are only n numbers between 1 and 2n, so we couldn't use the hypothesis to prove P(n + 1). We would have to prove case 3 by different means.
Prove by mathematic induction that all simple undirected graphs with 2 or more vertices have at least 2 vertices with the same degree.
This proof is very easy with the pigeon hole principle and 3 cases.
Case 1: the entire graph is connected.
Since it is a simple undirected connected graph, the range of possible degree values for a given vertex is 1 to n-1 for a graph with n vertices. Since there are n vertices and n-1 degree options for the vertices, there is at least a pair of vertices with the same degree. Ceiling(n/(n-1)) > 1.
Case 2: only one of the vertices is an isolated vertex.
Simply consider the connected subgraph of n-1 vertices. The range of possible degrees is 1 to n-2. Since there are n-1 vertices, the same thing holds as it did in case 1. Ceiling((n-1)/(n-2)) > 1
Case 3: two or more vertices are isolated vertices
Those multiple isolated vertices have the same degree, 0.
I don't like that we're given problems to use mathematic induction on when there exists are far more elegant solution. It can be annoying to tutor people on it because students tend to just want to do the easier proof in those cases even if it isn't the allowed method.
boviscopophobic
Member
While the pigeonhole proof is the nicest, I don't see the point in giving an induction proof that isn't using the induction hypothesis. You might as well just say "This direct proof is better so I'm doing that."
If you did wish to give a proof that used the induction hypothesis in a nontrivial way, here's how you could do it: Let A be a set of n+2 integers between 1 and 2n+2, and let B be the subset of A consisting of numbers <= 2n. If |B| >= n+1, then we are done by the induction hypothesis. The only case we need to worry about is that |B| = n and no elements of B divide each other. In this case 2n+2 is an element of A. If n+1 is in B, then we are done. If n+1 is not in B, then B union {n+1} contains two elements that divide each other, by the induction hypothesis. But this implies that B contains a divisor of n+1, which is thus a divisor of 2n+2, so we are done.
themagicalkitsune
Member
Yes in fact Hom(V,V) is isomorphic to Mat(n,F), the set of nxn matrices with entries in the base field. That's what I'm going to try next, thanks!
Hey, I just started learning linear algebra and I have a question in regards to identifying if a homogeneous system has one or infinite solutions. Today we were told that in homogeneous systems where there are more unknowns than equations, there will be infinite solutions.
The problem is, there wasn't any explanation as to what happens in regards to all zero rows. Do I count those when determining if a system of linear equations has infinite solutions.
An example (just random numbers tossed in RREF but I'm more curious about the theory):
_ _
| 1 0 0 0 | 0 |
| 0 1 0 0 | 0 |
| 0 0 1 1 | 0 |
|_0 0 0 0 | 0_|
Would you include the bottom row in determining the number of solutions, or would you ignore it?
Also, is there any easy way to determine if there would be 1 or infinite solutions in non-homogenous systems? I'm confused in regards to this as the prof brushed it off so quickly. He apparently is quizzing us on this tomorrow but even after studying it wasn't mentioned in the textbook.
The problem is, there wasn't any explanation as to what happens in regards to all zero rows. Do I count those when determining if a system of linear equations has infinite solutions.
An example (just random numbers tossed in RREF but I'm more curious about the theory):
_ _
| 1 0 0 0 | 0 |
| 0 1 0 0 | 0 |
| 0 0 1 1 | 0 |
|_0 0 0 0 | 0_|
Would you include the bottom row in determining the number of solutions, or would you ignore it?
Also, is there any easy way to determine if there would be 1 or infinite solutions in non-homogenous systems? I'm confused in regards to this as the prof brushed it off so quickly. He apparently is quizzing us on this tomorrow but even after studying it wasn't mentioned in the textbook.
boviscopophobic
Member
The all-zero row corresponds to the tautological equation 0 = 0, which doesn't tell you anything about the unknowns. So for the purposes of analyzing your system of equations, the zero rows might as well not even be there.
As for non-homogeneous systems, if the system is consistent (has at least one solution), then the number of solutions is the same as the homogeneous system that you would get by setting all the constant terms to 0. This is because: (a) if v is a solution of the nonhomogeneous system and w is a solution of the corresponding homogeneous system, then v + w is also a solution of the nonhomogeneous system; (b) if u and v are two solutions of the nonhomogeneous system, then u - v is a solution of the corresponding homogeneous system (can you prove these?)
the-pi-guy
Member
Does anyone have any good sources for Discrete Math?
I think I understand the stuff, but then when it comes to the assignments I am having lots of difficulties.
I think I understand the stuff, but then when it comes to the assignments I am having lots of difficulties.
the-pi-guy
Member
Sorry for the double post.
Are these two equivalent?
-For some x P(x) or for some x Q(x)
-For some x (P(x) or Q(x))
Are these two equivalent?
-For some x P(x) or for some x Q(x)
-For some x (P(x) or Q(x))
D
Deleted member 231381
Unconfirmed Member
The easiest way is to just test the permutations. Suppose there is some x such that P(x) and some x such that Q(x). The first statement is true, the second statement is also true (assuming the or is not an exclusive or). Suppose there is some x such that P(x), but no x such that Q(x). The first statement is true, the second statement is true. The same holds for there being Q(x) but no P(x). Finally, suppose there is no x such that either P(x) or Q(x). Then both statements are false. For all possible permutations, both statements are equivalent.
Note that this does assume the ors are inclusive, not exclusive.
This wouldn't work if it was and instead of or in this case - there might be some x such that P(x), and some different x such that Q(x), but no x such that both P(x) and Q(x).
the-pi-guy
Member
Thanks I think I got it.
MoG EclipsE
Member
Having trouble understanding how to number degrees of freedom at joints. (Ignore the data given below, it's used for another part of the question...)
Can anyone help with this?
I am thinking it's 4, 1, 2, 3 because the fixed end has less freedom?
Thanks!
Can anyone help with this?
I am thinking it's 4, 1, 2, 3 because the fixed end has less freedom?
Thanks!
Having trouble understanding how to number degrees of freedom at joints. (Ignore the data given below, it's used for another part of the question...)
Can anyone help with this?
I am thinking it's 4, 1, 2, 3 because the fixed end has less freedom?
Thanks!
It's a strange question to me, cause really the order of dofs can be anything you'd like. (It's true that solving the matrix equation/simultaneous equations can be easier if ordered right, but I'm not sure how your professor wants to order them.)
stanley1993
Neo Member
EDIT: I got it.
Trying to find derivative of the following function:
y = sin(pi x)^2
The answer is provided as:
2pi^2 x cos(pi^2 x^2)
through the use of chain rule.
Where I am stuck is with derivative of constant. Shouldn't taking derivative of pi make the whole function 0?
I'm guessing they simplified the function to sin(pi^2 x^2)
And then they applied the power rule wherein it becomes cos(pi^2 x^2)(2x pi^2). So they are not taking derivative of pi at all and hence the result. Why is that permissible? What am I missing?
y = sin(pi x)^2
The answer is provided as:
2pi^2 x cos(pi^2 x^2)
through the use of chain rule.
Where I am stuck is with derivative of constant. Shouldn't taking derivative of pi make the whole function 0?
I'm guessing they simplified the function to sin(pi^2 x^2)
And then they applied the power rule wherein it becomes cos(pi^2 x^2)(2x pi^2). So they are not taking derivative of pi at all and hence the result. Why is that permissible? What am I missing?
Omega Ultimus
Member
Going over your original post made me want to go over predicate logic again, as I noticed the same structure in your statements that are also found in simple logic-based ones. I may be wrong, but if you translate your sentence into a predicate logic form, you can show that they are equivalent through this sort of algebra:
∃x(A(x) ∨ B(x)) ≡ (∃x A(x) ∨ ∃x B(x))
http://www3.cs.stonybrook.edu/~cse371/PredLogicIntro2.pdf
If anyone notices I made a mistake, please point it out! This is as much a refresher for me as it is a pointer for the pi-guy (if they're interested in receiving one).
GAF Please! 
I haven't done math in a while, and this is not nearly as complicated as other things I see posted here, but I need to do some for a Quantitative Analysis for Business class I am in. I've messed this formula up twice and can't figure out where I am screwing up. I think it's in the order of operations, but I don't know. I am not good at math.
This is the forumla:
And here is my work, the values are correct, but the answer is wrong:
Q = √2DCs / (Ch (1-D/P))
D = 5,200,000
P = 88,400,000
Ch = 10 (.04 * 250)
Cs = 500
2 * 5,200,000 = 10,4000,000 * 500 = 5,200,000,000
1 - 5,200,000 / 88,400,000 = .94118
10 * .94118 = 9.4118
5,200,000,000 / 9.4118 = 48,941,360,000
√ 48,941,360,000 = 221,226.9423
Edit: Is Q = 23,505 the correct answer?
I haven't done math in a while, and this is not nearly as complicated as other things I see posted here, but I need to do some for a Quantitative Analysis for Business class I am in. I've messed this formula up twice and can't figure out where I am screwing up. I think it's in the order of operations, but I don't know. I am not good at math.
This is the forumla:
And here is my work, the values are correct, but the answer is wrong:
Q = √2DCs / (Ch (1-D/P))
D = 5,200,000
P = 88,400,000
Ch = 10 (.04 * 250)
Cs = 500
2 * 5,200,000 = 10,4000,000 * 500 = 5,200,000,000
1 - 5,200,000 / 88,400,000 = .94118
10 * .94118 = 9.4118
5,200,000,000 / 9.4118 = 48,941,360,000
√ 48,941,360,000 = 221,226.9423
Edit: Is Q = 23,505 the correct answer?
This step is incorrect (it seems like you multiplied by 9.4118 instead of dividing by it). Try it again, and you should get Q = 23505.319, approx.
Bottom of the page so probably will have to repost but I'm taking an Ordinary Differential Equations class. Can't figure out how to do the problems below:
I end up getting the integral of e^(-x^2) and I have no idea how to do that so I'm assuming I'm assuming I'm just going about it the wrong way.
I assumed I needed to find some integrating factor that would make this equation exact but I can't figure out how to get to cancel all nice like the usual problems.
I end up getting the integral of e^(-x^2) and I have no idea how to do that so I'm assuming I'm assuming I'm just going about it the wrong way.
I assumed I needed to find some integrating factor that would make this equation exact but I can't figure out how to get to cancel all nice like the usual problems.
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