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where the length z of the third side has been replaced by the vector sum x + y. When x and y are real numbers, they can be viewed as vectors in R1, and the triangle inequality expresses a relationship between absolute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 everyone,
I have a question on Rudin's proof of Theorem 1.33 part e. Here he prove the following statement:
The absolute value of z+w is equal or smaller than the absolute value of z plus the absolute value of w -Yes, is the triangle inequality, where z and w are both complex numbers-...
Hi everyone,
I am trying to solve an optimization problem using fmincon in Matlab with a nonlinear inequality restriction. Part of the objective function is undefined if this nonlinear inequality is violated. I also set up lower and upper bounds for fmincon. I use the "interior-point"...
Dear friends,
I am interesting to find some functions g satisfying the following convolution inequality
(g\astv)(t)\leqv(t)
for any positive function v\inL^{1}[0,T] and * denotes the convolution between g and v.
Hello gurus,
I've been trying to prove the following inequality for several days:
\int_1^\infty \frac{\exp\left(-\frac{(x-1)^2}{2a^2}\right)}{x}dx > \ln(1+a)\quad \forall a>0.
I've demonstrated by simulations that this inequality holds. I‘ve also proved that this inequality holds for large...
1. Homework Statement
f(x)=(2x^3+2x^2-5x-2) / 2(x^(2)-1)
f''(x)=(-12x^5-24x^3+36x)/(4x^8-16x^6+24x^4-16x^2+4)
Find the intervals where f is concave up.
2. The attempt at a solution (I am having trouble interpreting the results at the end or if I've made a mistake somewhere):
Attempt at...
Homework Statement
I am asked to prove:
2n < (n+1)! , where n≥2
The Attempt at a Solution
Base step: set n=2, then test 22 < (2+1)!
22 = 4
(2+1)!= 3! = 3(2)(1) = 6
so 4 < 6 , which is true.
Induction hypothesis is 2k < (k+1)!
Using this, prove 2(k+1) < [(k+1)+1]! = (k+2)!
Attempt to...
Homework Statement
Let a and b be real numbers
a. The condition “a + b = 0” is ...for the condition “a = 0 and b = 0”
b. The condition “a + b > 0” is ...for the condition “a > 0 and b > 0”
c. The condition ab = 0 is .... for the condition a = b = 0
d. The proposition “ a + b > 2 and ab >...
Hello everybody,
I've been trying to understand the CHSH proof as it is listed on Wikipedia:
http://en.wikipedia.org/wiki/CHSH_inequality
I got to this without any problem:
E(a, b) - E(a, b^\prime) = \int \underline {A}(a, \lambda)\underline {B}(b, \lambda)[1 \pm \underline {A}(a^\prime...
Copied from the OWS May Day protest thread...
This is a very common argument on here and I've seen it in other contexts as well. Usually it is in the context of a larger discussion about income inequality, but it is treated as a self-evident, throw-away claim that doesn't ever get...
Homework Statement
Let p > 1 and x > y > 0 Use the MVT to prove the inequality
py^(p-1)[x-y] =< x^p - y^p =< px^(p-1)[x-y]
The Attempt at a Solution
The only way i only how to use the MVT is where i already have the function. Do you have to define the function from the problem...
5. Suppose that x, y and z are positive real numbers such that xyz = 1.
(a) Prove that
27 \leq(1 + x + y)^{2} + (1 + y + z)^{2} + (1 + z + x)^{2}
with equality if and only if x = y = z = 1.
(b) Prove that
(1 + x + y)^{2} + (1 + y + z)^{2} + (1 + z + x)^{2} \leq 3(x + y + z)^{2}
with...
Homework Statement
Prove: If a>b and c>d, then a+c>b+d
Hint: (a-b)+(c-d)=(a+c)-(b+d)>0
Homework Equations
The Attempt at a Solution
How to use the hint to prove the inequality?
My method, not sure it's right.
Given c>d, c-d>0
Given a>b => a+(c-d)>b
Thus a+c>b+d
Homework Statement
{(x+y+z)^3-2(x+y+z)(x^2+y^2+z^2)}/xyz ≤ 9
Homework Equations
AM-GM inequality x+y+z ≥ 3(√xyz)(cube root) and xy+yz+zx ≥ 3√(xyz)^2(cube root)
The Attempt at a Solution
This is my attempt but I don't know if I am using the AM-GM inequality correctly...
Good day, I have this problem that appeared in some practical problem that I'm working on.
I basically want to find the boundaries of a,b,c,d for which the following inequality is satisfied, if a,b,c,d \in ℝ^+ and the inequality is:
-2 \cdot d + c - a \cdot (c \cdot d)^2 + a \cdot c +...
Homework Statement
Suppose \int_{-\infty}^{\infty}t|f(t)|dt < K
Using Cauchy-Schwartz Inequality, show that \int_{a}^{b} \leq K^{2}(log(b)-log(a))
Homework Equations
Cauchy Schwartz: |(a,b)| \leq ||a|| \cdot ||b||
The Attempt at a Solution
Taking CS on L^{2} gives us...
Homework Statement
\frac{1^2*3^2*5^2...(2n-1)^2}{2^2*4^2*6^2...(2n)^2}<\frac{1}{2n+1}
Edit: Must be proven without using induction.
Homework Equations
The Attempt at a Solution
I understand the LHS is the same thing as
\frac{(2n-1)!}{(2n)!}
And (2n)! = k!2^k & (2n-1)! =...
Homework Statement
If 0 <= A <= B, prove that: A(B-A) <= (B/2)^2
Homework Equations
-
The Attempt at a Solution
I've been blindly rearranging the terms trying to see a way to prove this but due to my complete lack of experience in proofs, I'm hoping someone here can give a little...
EDIT: Turns out, the solution to my question is related to the determinant of a positive definite quadratic form.
This is more or less straight from Landau's Statistical Physics Part 1 (3rd edition), Chapter 21.
I don't understand how the inequality/condition (the last equation in this post)...
I want to prove
cos2(x)/(n2 + 1) ≤ 1/(n2 + 1)
I know this is an obvious inequality but I want to know if my reasoning is correct.
For the expression cos2(x)/(n2 + 1) to be as large as possible the numerator must → ∞ but cos2(x) is bounded above by 1.
cos2(x) = 1 for x = 2∏k where k ≥1...
I just got Spivak's calculus today, and I'm already stuck on the prologue problems:
1. The problem
Find all x for which (x-1)(x-3)>0
2. The attempt at a solution
We know that if ab>0, then either a>0 and b>0, or a<0 and b<0.
Thus, if a=(x-1) and b=(x-3), then either (x-1)>0 and...
Obviously a violation of the CHSH inequality means that local realistic theories are untenable.
If we sent two entangled photons towards detectors (far enough away that for information to travel, you'd require it to go faster than light). One reaches a detector before the other, so...
Proving the "triangle inequality" property of the distance between sets
Here's the problem and how far I've gotten on it:
If you are unfamiliar with that notation, S(A, B) = (A \ B) U (B \ A), which is the symmetric difference.
And D(A, B) = m^*(S(A, B)), which is the outer measure of...
34.
Turner's syndrome is a rare chromosomal disorder in which girls have only one X chromosome. It affects about 1 in 2000 girls in the United States. About 1 in 10 girls with Turner's syndrome also suffer from an abnormal narrowing of the aorta.
a. In a group of 4000 girls, what is the...
Homework Statement
Solve
|3x-7|-|x-8|>4
The Attempt at a Solution
so i made columns... and using the columns i made a number line..
7/3 on the left as a point, with a column on its left, and 8 with a column on its right and sharing a coumn in the middle with 7/3
so i have...
The problem
Given the Schwarz inequality, x_{1}y_{1} + x_{2}y_{2} \leq \sqrt{x_{1}^{2} + x_{2}^{2}} \sqrt{y_{1}^{2} + y_{2}^{2}}, prove that if x_{1} = \lambda y_{1} and x_{2} = \lambda y_{2} for some number \lambda \geq 0, then equality holds. Prove the same thing if y_{1} = y_{2} = 0. Now...
Hi,
Given $ A+B+C=\pi$, I need to prove $ cosA+cosB+cosC\leq \frac{3}{2}$.
I wish to ask if my following reasoning is correct.
First, I think of the case where A and B are acute angles, then I can use the Jensen's Inequality to show that the following is true.
$ cos\frac{A+B}{2}\geq...
I am having trouble proving these. I cannot figure out how to get to the conclusion. Here is my attempt. The stuff in red is just side work and is not part of the proof. I always get stuck on these types of problems, can someone offer some tips on how to approach these kind of problems in...
Homework Statement
Show that if |z| = 10 then 497 ≤ |z^{3} + 5iz^{2} − 3| ≤ 1503.
The Attempt at a Solution
I'm not an entirely sure how to begin this one, or if what I'm doing is correct.
If I sub in |z| = 10 into the equation; |1000 + 500i - 3| = 997 +500i
Then the modulus of...
Homework Statement
By expanding (x-y)^2, prove that x^2 +y^2 ≥ 2xy for all real numbers x & y.
Homework Equations
The Attempt at a Solution
expanding (x-y)^2
x^2 - 2xy + y^2= 0
Hence, x^2 + y^2 = 2xy
But where does the ≥ come into it? and why?
when you put values in...
inequality... how does this make sense??
Homework Statement
Solve (x-1)(x-2)<0
Homework Equations
The Attempt at a Solution
Given this is a parabola
graphical solution
cuts the x-axis at 1 & 2
therefore sltn...
1<x<2
However, in my textbook the answer says...
Hi just a quick question I was curious about.
Im not sure if the results from CERN about the faster than light neutrino have been verified, but given that this is true...
as I understand it bell's inequality assumes
1. the reality of the external world, independent of us "observers".
2...
I am trying to show $|(n+z)^2|\leq (n -|z|)^2$ where is complex
$|(n+z)^2| = |n^2 + 2nz + z^2| \leq n^2 + 2n|z| + |z|^2$ But I can't figure out the connection for the final piece.
Here's a nice problem.
Suppose $a_1,a_2,...,a_n$ are postive real numbers satisfying \(a_1\cdot a_2\cdots a_n=1\). Show that $(a_1+1)(a_2+1)\cdots(a_n+1)\geq2^n$.
Homework Statement
Show that if z_1,z_2 \in \mathbb{C} then |z_1+z_2| \leq |z_1| + |z_2|
Homework Equations
Above.
The Attempt at a Solution
I tried by explicit calculation, with obvious notation for a,b and c: my frist claim is not that the triangle inequality holds, just that...
As I can see from the formula of cauchy inequality:
(a1^2+a2^2+...+an^2)^1/2 . (b1^2+b2^2+...+bn)^1/2 >= a1b1+a2b2 + ... + anbn
Can I conclude from the above formula that:
(a1+a2+...+an)^1/2 . (b1+b2+...+bn)^1/2 >= (a1b1)^1/2 + (a2b2)^1/2 +...+ (anbn)^1/2
by setting a1,...,an =...
I have an inequality and tried to solve it and reached the following:
Original question: Prove (1/a - 1)(1/b - 1)(1/c - 1) >= 8 when a+b+c = 1 and a,b,c positive
After expanding and some eliminations,
I still need to prove 1/a + 1/b + 1/c -1 >= 8
Any suggestion how to solve it?
Homework Statement
Digital filter analysis - this is just one part of a multi-part question I can't move forward with. It's supposed to be an auxilliary question and isn't the "meat" of the problem.
Find b, such that maximum of the magnitude of the frequency response function...
Hello to everyone. This is my first time here so I hope I will not cause any unwanted trouble.
Straight to the problem. I have one inequality for which I would like to prove, but I do not know how. The inequality has the following form:-
(1+a)q < q/(1-a), where a < 1 and q can be any positive...
Homework Statement
The problem is:
for all 0≤a≤1
so i need to find the domain
Homework Equations
N/A
The Attempt at a Solution
I tried it like this:
yet my solution is wrong,i am not so sure why.
wolfram gives me this;