What is Inequality: Definition and 1000 Discussions
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but some authors, especially those writing about elementary geometry, will exclude this possibility, thus leaving out the possibility of equality. If x, y, and z are the lengths of the sides of the triangle, with no side being greater than z, then the triangle inequality states that
z
≤
x
+
y
,
{\displaystyle z\leq x+y,}
with equality only in the degenerate case of a triangle with zero area.
In Euclidean geometry and some other geometries, the triangle inequality is a theorem about distances, and it is written using vectors and vector lengths (norms):
‖
x
+
y
‖
≤
‖
x
‖
+
‖
y
‖
,
{\displaystyle \|\mathbf {x} +\mathbf {y} \|\leq \|\mathbf {x} \|+\|\mathbf {y} \|,}
where the length z of the third side has been replaced by the vector sum x + y. When x and y are real numbers, they can be viewed as vectors in R1, and the triangle inequality expresses a relationship between absolute values.
In Euclidean geometry, for right triangles the triangle inequality is a consequence of the Pythagorean theorem, and for general triangles, a consequence of the law of cosines, although it may be proven without these theorems. The inequality can be viewed intuitively in either R2 or R3. The figure at the right shows three examples beginning with clear inequality (top) and approaching equality (bottom). In the Euclidean case, equality occurs only if the triangle has a 180° angle and two 0° angles, making the three vertices collinear, as shown in the bottom example. Thus, in Euclidean geometry, the shortest distance between two points is a straight line.
In spherical geometry, the shortest distance between two points is an arc of a great circle, but the triangle inequality holds provided the restriction is made that the distance between two points on a sphere is the length of a minor spherical line segment (that is, one with central angle in [0, π]) with those endpoints.The triangle inequality is a defining property of norms and measures of distance. This property must be established as a theorem for any function proposed for such purposes for each particular space: for example, spaces such as the real numbers, Euclidean spaces, the Lp spaces (p ≥ 1), and inner product spaces.
Hi, so here is my question that I am totally stumped on.
for all real values of x and y, show that |x|+|y|≥ √(x^2+y^2 )
and find the real values of x and y in which equality holds.
I sort of thought I could do the second part, but it confuses me with two pronumerals and how to get rid...
Deduce from the simple estimate that if 1<\sqrt{3}<2, then 6<3^{\sqrt{3}}<7.
Hi members of the forum,
This problem says the resulting inequality may be deduced from the simple estimate, but I was unable to do so; could anyone shed some light on how to deduce the intended result?
Thanks in...
Let C[-pi,pi] be the set of continuous function from [-pi,pi] to C. Endow this with usual inner product (<f,g>= integral from -pi to pi of f multiplied by g conjugate, and let ||.|| be the corresponding norm).
Let h(n) be Fourier coefficent of fNow, |h(n)|<_ 1/2pi( ||f||.||e^int||) by schwarz...
Homework Statement
Find constraints on a,b,c \in \mathbb{R} such that \forall w_1,w_2,w_3 \in \mathbb{C} ,
(1) x = |w_1|^2(1-c) + a|w_2|^2 + c|w_1+w_3|^2 + |w_3|^2(b-c) \ge 0 and
(2) x=0 \Rightarrow w_1=w_2=w_3=0 .Homework Equations
The Attempt at a Solution
I believe the solution is...
Homework Statement
|4 + 2r - r^2| <1
Homework Equations
4 + 2r - r^2 = (r - (1+ √5) ) (r - (1 - √5))
The Attempt at a Solution
I tried to use the roots but no use. How should I proceed?
Homework Statement
The task is to find all solutions of the following inequality:
x+3^x <4
But I was trying to find a solution for this problem in general:
x+a^x < b
Homework Equations
n/a
The Attempt at a Solution
a^x < b-x
\text{log}_a(a^x) < \text{log}_a...
Homework Statement
Show that 2-norm is less equal to 1-norm
But I've found this proof
http://img825.imageshack.us/img825/5451/capturaklt.jpg
Which basically shows that if p=1 and q=2 then 1-norm is less equal than 2-norm, i.e. the opposite hypothesis
Homework Equations
NoneThe Attempt at...
Homework Statement
Question 1:
You find an old map revealing a treasure hidden on a small island. The treasure was buried in the following way: the island has one tree and two rocks, one small one and one large one.
Walk from the tree to the small rock, turn 90 to the left and walk the same...
I have proven to n=3 an inequality that seems useful. x1n+...+xnn≥nx1...xn for all positive x.
I'm sure this has been proven before. I'm not quite sure how to extend it from n=3 to for all n. I'm thinking induction, but that has proven challenging. Any hints?
Homework Statement
Prove that:
Homework Equations
\sum_{k=1}^{n}x_{k}^{2}\geq \frac{1}{n}\left ( \sum_{k=1}^{n}x_{k} \right )^{2}
The Attempt at a Solution
I am not sure what to do to be honest. But it looks like the Cauchy–Schwarz inequality to me.
Homework Statement
these two functions will give the same Fourier series? because when I write the graph they look the same?
Homework Equations
The Attempt at a Solution
in the picture
thank you
Homework Statement
Question 3 here: http://www.stat.washington.edu/peter/395/samplemidterm.pdf
Solution to it here: http://www.stat.washington.edu/peter/395/prmt.sln/sln.html
By the way, I could use help soon since this is the practice exam for an exam I'm taking 7 hours from now...
Homework Statement
Let E have finite outer measure. Show that if E is not measurable, then there is an open set O containing E that has finite measure and for which
m*(O~E) > m*(O) - m*(E)
Homework Equations
The Attempt at a Solution
This is what I did...
m^*(O) = m^*((O \cap E^c) \cup...
Homework Statement
3 - 3^(x+1) < 5^x - 15^x
3(1-3^x) < 5^x(1-3^x)
Do I have to impose 1-3^x > 0 ?
It results x<0 and x>log(5,3) but book has written 0 < x < log(5,3) where did I wrong ?
Homework Statement
I need some help setting up this inequality:
How accurate do the sides of a cube have to be measured if the volume of the cube has to be within 1% of 216 cm^3
Not very good with word problems and for some reason this course never deals with them until now? And this is...
Homework Statement
The range of k for which the inequality ##k\cos^2x-k\cos x+1≥0## for all x, is
a. k<-1/2
b. k>4
c. -1/2≤k≤4
d. -1/2≤k≤2Homework Equations
The Attempt at a Solution
I am not sure about how to begin with this. This seems to me a quadratic in cos(x) and here, the discriminant...
Problem:
Let x and y be positive real numbers satisfying the inequality $\displaystyle x^3+y^3\le x-y$.
Prove that $\displaystyle x^2+y^2\le 1$ .
Hi all, I'm at my wit's end to prove the question as stated above, and I know it's obvious that $\displaystyle x-y>0$ and $\displaystyle...
Homework Statement
Let ##a## and ##b## real numbers such that ##a>b>0##.
Determine the least possible value of ##a+ \frac{1}{b(a-b)}##
I took this example from page 3 of this paper
Homework Equations
In the article previously linked, explaining the example, the author writes...
Homework Statement
If g(x)\ge 0, then for any constant ##c>0, r>0##:
P(g(X)\ge c)\le \frac{E((g(X))^r)}{c^r}
Homework Equations
I know that E(g(X))=\int_0^\infty g(x)f(x)dx if g(x)\ge 0 where ##f(x)## is the pdf of ##X##.
The Attempt at a Solution
I tried following a similar...
I imagine that some topics and questions keep reappearing since it is hard to track through all past posts even with the query tool. So apologies if this has been covered before (as it probably has). I just want to check my intuitive understanding of the Bell experiment, having heard an...
Let f(x) \in C[a,b] and let f(x)>0 on [a,b]. Prove that
\exp \Big(\frac{1}{b-a}\int_a^b \ln f(x) dx \Big)\leq \frac{1}{b-a}\int_a^b f(x) dx
I have learned Gronwall's Inequality and Jensen's Inequality(and inequality deduced from it like Cauchy Schwarz Inequality) but i couldn't use them to...
Hey! I have this: 2(√(1-a^2 ))+ 2a
How to determine the maximum value of this?
I think good for this is Inequality of arithmetic and geometric means, but I don't know how use this, because I don't calculate with this yet.
So, have you got any ideas?
Poor Czech Numeriprimi... If you...
Hi,
can someone give me a refference for the FKG inequality?
that is,
let f,g be two nondecreasing and nonegative measurable functions on a probability space.Then
∫fgdμ≥(∫fdμ)(∫gdμ)
Hi everybody,
while doing a complex analysis exercise, i came to a strange inequality which i don't know how to interpretate. Suppose you have a sequence $\{a_j\}$ of positive real number. Let $\rho$ a positive real number. The inequality i found after some calculation is...
Hi all, I've been having a hard time trying to solve the following inequality:
Prove that $\displaystyle \left(\log_{24}(48) \right)^2+\displaystyle \left(\log_{12}(54) \right)^2 >4$
I've tried to change the bases to base-10 log and relating all the figures (12, 24, 48, and 54) in terms of 2...
1. Another approach to the proof of the Cauchy-Shwarz Inequality is suggested by figure 16 (sorry, I don't have the image), which shows that, in ℝ2 or ℝ3, llproj_u{}vll ≤ llvll. Show that this inequality is equivalent to the Cauchy-Schwartz Inequality.
2. Cauchy-Schawrtz Inequality: lu • vl...
Homework Statement
For every x in the interval [0,1] show that:j
\frac{1}{4}x+1\leq\sqrt[3]{1+x}\leq\frac{1}{3}x+1
The Attempt at a Solution
Well i subtracted 1 from all sides and divided by x and I got:
\frac{1}{4}\leq\frac{\sqrt[3]{1+x}-1}{x}\leq\frac{1}{3}
But now I need to find a...
Homework Statement
\frac{2^{x+1}-3}{2^{x}-4}\leq1
Homework Equations
The Attempt at a Solution
When I go through it, I keep getting 2^{x}\leq-1
I don't think the answer is suppose to be complex... let me show you my work in a file.
It's extremely embarrassing that I don't...
Hello there,
Im studying QM with Shankar's book.
I'm wrestling myself trough the linear algebra now and I have some questiosn.
Let me start with this one:
I have absolutely no idea where this is coming from or what does it mean.
I don't know how to multiply a ket with an inner...
I am trying to solve the following inequality:
x3 <= 7x3 - 24x2 + 6x - 10
I have worked it out as follows:
0 <= 6x3 - 24x2 + 6x - 10
10 <= 6x3 - 24x2 + 6x
10 <= 6x(x2 - 4x + 1)
At this point, I'm not sure how to proceed and I'm not sure if the factoring on the last step was helpful. Any...
Homework Statement
Solve the inequality -9 < 1/x
A simple inequality, I can see the solution is just x < -1/9 but I can't prove it at all.
The Attempt at a Solution
-9 < 1/x
-9x < 1
x > -1/9
Any helpful rules I am forgetting about inequalities? This was a problem in a review...
Hello I have a worked example where I have to maximize a function with an inequality constraint. The problem is worked out below.
https://zgqqmw.sn2.livefilestore.com/y1pLc13HVWpA9dATZEzikySeSMBN2hn1mJCw71rJ5vvUJcr9W7KBPFkOz7HQEppa6EPbLi5yyAwDagh3ezF_7eyVL6tBK7q6ise/maxProbem.png?psid=1
I...
Homework Statement
If n is a positive integer, prove that 2^n > 1+n\sqrt{2^{n-1}}
Homework Equations
The Attempt at a Solution
I am thinking of applying AM GM HM inequality. But which numbers should I take to arrive at this inequality?
Solve the rational inequality (a-5)/(a+2) < -1.This is what I got so far:
a-5/a+2 = -1
a-5= -a-2
0= -2a+3
subtract 3 from both sides:
-3=-2a
Divide by -2
3/2=a
I know that the answer is (-2, 3/2), but I'm not sure where the -2 in the answer comes from. Thanks!
Homework Statement
Let V be a real inner product space, and let v1, v2, ... , vk be a set of orthonormal vectors.
Prove
Ʃ (from j=1 to k)|<x,vj><y,vj>| ≤ ||x|| ||y||
When is there equality?
Homework Equations
The Attempt at a Solution
I've tried using the two inequalities given to us in...
Homework Statement
If a,b,c are the positive real numbers, prove that a^2(1+b^2)+b^2(1+c^2)+c^2(1+a^2) \geq 6abc
Homework Equations
The Attempt at a Solution
With a little simplification L.H.S = (a^2+b^2+c^2)+(a^2b^2+b^2c^2+c^2a^2)
Using A.M>=G.M
\dfrac{a^2+b^2+c^2}{3} \geq...
Prove that if ##m > 1## such that there exists a ##c > 1## that satisfies
$$cm < m^c$$
then for any ##k > c##
$$km < m^k$$
holds. Prove this without using logarithms or exponents or calculus. Basically using the properties of real numbers to prove this.
One attempt I have tried, but didn't...
Homework Statement
Two numbers x and y are selected from a closed interval [0,4]. To find the probability that the two numbers satisfies the condition that y^{2}\leq x.
2. The attempt at a solution
Don't have any idea
Homework Statement
Ax ≤ b, assuming A is nxn and solution exists
Homework Equations
The Attempt at a Solution
I don't know of any concrete methods offhand. A grad student suggested rearranging it to:
Ax - b ≤ 0, zero vector
Then I don't know where to go from here. I was...
I'm trying to understand a process called order finding as I need to know it for Shor's algorithm in quantum computing.
The process is like this:
For two positive integers x and N, with no common factors and x < N, the order of x modulo N is defined to be the least positive integer, r...