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where the length z of the third side has been replaced by the vector sum x + y. When x and y are real numbers, they can be viewed as vectors in R1, and the triangle inequality expresses a relationship between absolute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.
Homework Statement
I am attempting to show that -x \leq sin(x) \leq x for x>0 and thus \int^1_0 nxsin(\frac{1}{nx})dx converges to 1.
Homework Equations
I know that I need to use the fact that I have shown that the limit as T tends to infinity of \int^T_1 \frac{cos(x)}{\sqrt{x}}dx...
I want to find value for m for which:
4m2 - 12m > 0
Say I do this algebraically:
4m(m-3) > 0
so m > 0 or m > 3
The answer however is 0 < m and m > 3, I know this as a fact as I have looked graphically.
So, my question is, when done algebraically, how do I get 0 < m instead of m...
Homework Statement
Show from definition that if f is measurable on [a,b], with m<=f(x)<=M for all x then its lebesgue integral, I, satisfies
m(b-a)<=I<=M(b-a)
Homework Equations
The Attempt at a Solution
I know that the definition is that f:[a,b]->R is measurable if for each t...
Homework Statement
Prove the Triangle Inequality Theorum using the coordinate system.
Homework Equations
The corners of the triangles will be at (x1,y1), (x2, y2), (x3,y3)
The Attempt at a Solution
The proof that I know is proving that |x+y|<=|x|+|y|:
-|x|<x<|x|, and...
Hi I'm in the process of trying to understand the proof to bessel's equality and inequality and I am stuck, I have got to the line
http://img141.imageshack.us/img141/396/besselsequality.jpg Uploaded with ImageShack.us
and I'm not entirely sure how it equates to the next line but according to...
My mathematical methods for theoretical physics course recently began looking at linear vector spaces. We defined the Banach and Hilbert Spaces and proved the Cauchy-Shwarz Inequality. There's one step in this proof that I can't really follow (in red):
consider: w=x+uy (i'll drop the...
Homework Statement
Hi Guys,
I try to find the range for parameters phi1 and phi2 were the autoregressive process below is stationary.
We have the process X(t)+phi1*X(t-1)+phi2X(t-2)=Epsilon(t) (1)
Homework Equations
We get the characteristic polynomial F(z)=z^2+phi1*z+phi2 (2)
The...
I have the following question on metric spaces
Let (X,d) be a metric space and x1,x2,...,xn ∈ X. Show that
d(x1, xn) ≤ d(x1, x2) + · · · + d(xn−1, xn2 ),
and
d(x1, x3) ≥ |d(x1, x2) − d(x2, x3)|.
So the first part is simply a statement of the triangle inequality. However, the metric...
Homework Statement
Prove the following inequality:
\frac{1}{6}\leq\int_{R}\frac{1}{y^{2}+x+1}\chi_{B}(x,y)dxdy\leq\frac{1}{2}
where B={(x,y)|0\leq (x)\leq (y)\leq1} and R=[0,1]x[0,1]
EDIT: The B region should be 0 less than or equal to x less than or equal to y less than or equal to 1...
I am having a bit of a problem with Chebychev's inequality which is:
P(|X-\mu |\geq \alpha )\leq \frac{\mathrm{Var}(X)}{\alpha ^2}
For a positive \alpha. Here X denotes a stochastic variable with mean \mu and finite variance. I am asked to give a direct proof of this result, using the...
Homework Statement
Show that a2+b2 =>2ab, and hence, if x+y+z=c, show that x2+y2+z2 => 1/3 c2
Homework Equations
The Attempt at a Solution
How to prove this when we only have unknowns? The only thing i can think of for the first one is a (a+b)2= a2+b2 +2ab, but how to prove that a2+b2 =>2ab...
Homework Statement
Given http://www.mathhelpforum.com/math-help/attachments/f33/20928d1298610998-function-msp281219ebge8he857gc6900005ba9285dff0f5h79.gif , find the values of ''a'' for which the value of the function f(x) <= 25/2.
The answer is a<= 1/2.
Homework Equations
The Attempt at a...
Hi there,
I am trying to prove the following. For any random vectors X,Y,Z,W in \mathbb{R}^d and deterministic d\times d matrices A,B the covariance
\operatorname{\mathbb{C}ov}\left(X^TAY;Z^TBW\right)
can in some way be bounded by the covariance...
Homework Statement
The problem is Excercise 5. in page 88 of Folland's "real analysis: modern techniques and their applications", 2nd edition, as the image below shows.
Homework Equations
As the hint indicates, we should use Excercise 4.
The Attempt at a Solution
From Excercise 4, if...
So I multiplied the heat equation by 2u, and put the substitution into the heat equation, and get 2uut-2uuxx=(u2)t=2(uux)x+2(ux)2.
I`m not sure where to go from there, I can integrate with respect to t, then I would have a u2 under the integral on the left side, but them I`m not sure where to...
How are these two equal??(equation, inequality)
I study discrete mathematics and we are doing combinations at the moment. There is this example in the book(Discrete Mathematics and Combinatorics p. 30 Ex. 1.43) where it states that the number of integer solutions for:
x1+x2+x3+x4+x5+x6<10...
hi, while trying to study complex analysis, i have a few problems.
i already know that in complex number system, it's impossible for any order relation to exist.
but i was confused to this fact when i saw the proof of triangle inequality.
;
Let z,w be complex numbers. Then, triangle...
Dear all,
Consider the system given by : http://www.freeimagehosting.net/image.php?53f7eed9ce.jpg
where we are trying to solve for s and gamma using Newton's method. It turns out to be a simple implementation. Now, what if we need to impose an inequality constraint on the solution s : one...
Homework Statement
If x,y\in R and x+y=1.then find max. and Min. value of (x^3+1)(y^3+1) (Without using calculus)
Homework Equations
here x+y=1 and (x^3+1)(y^3+1)
The Attempt at a Solution
I have done using Calculus...
Homework Statement
then what are the operations that maintain the Inequality and what are the operations that don't?
Homework Equations
The Attempt at a Solution
clearly addition and subtraction maintains it ,and so does multiplication and division by any number other than...
I'm trying to figure out how to prove this inequality:
I know it's true (by graphing), but what's an algebraic way to prove it?
1+\frac{1}{3x^2}< x \tan \frac{1}{x} < \frac{1}{\sqrt{1-\frac{2}{3x^2}}}
Thanks
Homework Statement
Prove without computation that 2<Integral[0,2] (1+x^3)<6
The Attempt at a Solution
I know there is a theorem which says that if a function is bounded by two constants, then the integral of the function is also bounded by the integrals of the two functions. However...
Homework Statement
Graph the following inequality in the complex plane: |1 - z| < 1
2. The attempt at a solution
In order to graph the inequality I need to get the left side in the form |z - ...|
|1 - z| < 1
|(-1)z + 1| < 1
|-1(z - 1)| < 1
|-1||z - 1| < 1
(1)|z - 1| < 1
|z - 1| < 1...
Homework Statement
Solve the Inequality:
(3x-7)/(x+2)<1
Homework Equations
The Attempt at a Solution
Cross Multiply: x+2>3x-7
Simplify: 9>2x
Simplify More: 9/2>x
My Answer: (-∞, 9/2)
I put this as my answer but the answer is really (-2, 9/2)
Can someone explain to me why this is? I know you...
Homework Statement
Let V be a vector space with inner product <x,y> and norm ||x|| = <x,x>^1/2.
Prove the Cauchy-Schwarz inequality <x,y> <= ||x|| ||y||.
Hint given in book: If x,y != 0, set c = 1/||x|| and d = 1/||y|| and use the fact that
||cx ± dy|| >= 0.
Here...
Homework Statement
Prove that ||a|-|b||\leq |a-b| for all a,b in the reals
Homework Equations
I know we have to use the triangle inequality, which states:
|a+b|\leq |a|+|b|.
Also, we proved in another problem that |b|\leq a iff -a\leqb\leqa
The Attempt at a Solution
Using the...
Question:
I need to prove this inequality:
Where x,y,x are non-negative and x+z<=2:
(x-2y+z)^2 >= 4xz -8y.
My attempt:
I thought maybe choosing x as 0 and z as 0 will and then solving for y... but that only yields y+2 >= 0, which isn't really a solution, since I can't choose numbers...
Homework Statement
Prove that if a,b > 1, then a+b < 1+ab
The Attempt at a Solution
Just want to know if this makes sense:
first let a+b < 1+ab become 1<(1+ab)/(a+b) ==> 0<(1+ab-(a+b))/(a+b).
Factoring the numerator: 0<(1-a+ab-b)/(a+b) ==> 0<(1-b)+a(b-1)/(a+b)
So the next...
Hi,
I was reading Heinz Pagels' description of the nail gun experiment in the chapter about
"Bell's Inequality" from his book, The Cosmic Code: Quantum Physics as the Langauge of
Nature, 1982, pp. 160-176. He describes the record of hits and misses after "turning
polarizer A clockwise by...
Got some exciting news from a PF Mentor:
And the actual paper:
http://iopscience.iop.org/1367-2630/12/12/123007
Violation of Leggett inequalities in orbital angular momentum subspaces
J Romero, J Leach, B Jack, S M Barnett, M J Padgett and S Franke-Arnold
J Romero et al 2010 New J...
Homework Statement
Let f(z) be an entire function such that |f(z)| less that or equal to R whenever R>0 and |z|=R.
(a)Show that f''(0)=0=f'''(0)=f''''(0)=...
(b)Show that f(0)=0.
(c) Give two examples of such a function f.
Homework Equations
The Attempt at a Solution...
Homework Statement
let n\inN To prove the following inequality
na^{n-1}(b-a) < b^{n} - a^{n} < nb^{n-1}(b-a)
0<a<b
Homework Equations
The Attempt at a Solution
Knowing that b^n - a^n = (b-a)(b^(n-1) + ab^(n-2) + ... + ba^(n-2) + a^(n-1) we can divide out (b-a) because b-a #...
Hello,
i've met during problem solving with inequality
\max\{A+B,C\}\le\max\{A,C\}+\max\{B,C\}
where A,B and C are real numbers. I don't know whether it holds, but I need to prove that.
Thanks for reply...
Homework Statement
find n0,c1,c2 for which the following is true:
c1 nb <=(n-a)b<=c2(n-a)b , for all n > n0Homework Equations
http://en.wikipedia.org/wiki/Binomial_theorem" ?The Attempt at a Solution
c1 nb <=(n-a)b<=c2(n-a)b
c1 nb <=nb-nb-1a+nb-2a2-...-ab<=c2nb
c1<=1-a/n + a2/n2-...
Homework Statement
For f nonnegative and continuous on [0,1], prove.
\left( \int f \right) ^2 < \int f^2
With the limits from 0 to 1.
Homework Equations
The Attempt at a Solution
I was trying to use Upper sums, i.e.
\inf \sum \Delta x_i M_i(f^2) = \inf \sum \Delta x_i...
Prove that in any triangle ABC with a sharp angle at the peak C apply inequality:(a^2+b^2)cos(α-β)<=2ab
Determine when equality occurs.
I tried to solve this problem... I proved that (a^2+b^2+c^2)^2/3 >= (4S(ABC))^2,
S(ABC) - area
but I don't know prove that (a^2+b^2)cos(α-β)<=2ab :(...
Is it true in general that:
|\int f(x)dx| < \int |f(x)|dx
Not sure if "Triangle Inequality" is the right word for that, but that seems to be what's involved.
Homework Statement
Prove that
2 \leq 1+ \sum(m=1 to n) 1/m! \leq 1 + \sum (m=1 to n) (1/(2^(m-1))) < 3
The Attempt at a Solution
I've proved by induction that 2m-1 \leq m!, so it just follows that
1 + (1/(2 ^ (m-1))) \geq 1 + (1/m!), and their sums are the same inequality...
suppose that g:[0,1] \rightarrow \re is continuous, g(0)=g(1)=0 and for every c \in (0,1), there is a k > 0 such that 0 < c-k < c < c+k < 1 and g(c)=\frac(1}{2} (g(c+k)+g(c-k)).
Prove that g(x) = 0 for all x \in [0,1] Hint: Consider sup{x \in [0,1] | f(x)=M } where M is maximum of f on [0,1]...
1. Suppose that f: (a,b) --> R is convex. Prove Jensen's inequality: if x1,...,xn\in(a,b) and c1,...,cn >= 0 s.t. \sum(c_j)f(x_j) >= f(\sum((c_j)(x_j))
both summations from j = 1 to n
2: Convex: whenever x1, x2 \in(a,b) and 0 <= c <= 1, we have cf(x1) + (1 + c)f(x2) >= f(cx1 + (1-c)x2)...
This was already posted by someone else but an answer wasn't received so I thought I'd repost. Any help is appreciated.
Homework Statement
Use the Maclaurin series for cosx and the Alternating Series Estimation Theorem to show that
\frac{1}{2} - \frac{x^2}{24} < \frac{1-cosx}{x^2} <...
Homework Statement
Use the Maclaurin series for cosx and the Alternating Series Estimation Theorem to show that
\frac{1}{2} - \frac{x^2}{24} < \frac{1-cosx}{x^2} < \frac{1}{2} Homework Equations
cosx = 1 - \frac{x^2}{2} + \frac{x^4}{4} - \cdot \cdot \cdot = \sum_{n=0}^\infty...
Homework Statement
Prove that if n is a natural number and is greater/equal to 4, then 2n<n!,
and show that 2n is less/equal to 2((n-1)!) follows.
The Attempt at a Solution
I'm thinking I just need to use induction on n for the first part, where I get the inequality
(n+1)! = n!*(n+1) >...
Homework Statement
d(x,y)=(a|x_1-y_1|^2+b|x_1-y_1||x_2-y_2|+c|x_2-y_2|^2)^{1/2}
where a>0, b>0, c>0 and 4ac-b^2<0
Show whether d(x,y) exhibits Triangle inequality?
Homework Equations
(M4) d(x,y) \leq d(x,z)+d(z,y) (for all x,y and z in X)
The Attempt at a Solution
I...
Homework Statement
Show that 1/2(1-cos1)\leq\int\intsinx/(1+(xy)4)dxdy\leq1 on the area 0\leqx\leq1, 0\leqy\leq1.
Homework Equations
Mean Value Inequality: m*A(D)\leq\int\intf(x,y)dA\leqM*A(D), where m is the minimum and M is the maximum on the interval.
The Attempt at a Solution...
Homework Statement
Use your knowledge of exponents to solve
\frac{1}{2^x} > \frac{1}{x^2}
Homework Equations
The Attempt at a Solution
x^2 > 2^x
Then I am stuck.
I know they intersect at x = 2.