Metric Spaces Calculator

Ask and get solution to your homeworkAsk now and get step-by-step solutions

Example

Created on 2024-06-20Asked by Liam Mitchell (Solvelet student)

Verify if the given set and distance function form a metric space. Consider the set

\mathbb{R}

with the distance function

d(x, y) = |x - y|

Solution

To verify if

(\mathbb{R}, d)

where

d(x, y) = |x - y|

forms a metric space, we need to check the following properties: 1. **Non-negativity**:

d(x, y) \geq 0 \quad \text{for all} \quad x, y \in \mathbb{R}.

Since

|x - y|

is always non-negative, this property holds. 2. **Identity of Indiscernibles**:

d(x, y) = 0 \quad \text{if and only if} \quad x = y.

x = y

, then

|x - y| = 0

. Conversely, if

|x - y| = 0

, then

x = y

. 3. **Symmetry**:

d(x, y) = d(y, x) \quad \text{for all} \quad x, y \in \mathbb{R}.

Since

|x - y| = |y - x|

, this property holds. 4. **Triangle Inequality**:

d(x, z) \leq d(x, y) + d(y, z) \quad \text{for all} \quad x, y, z \in \mathbb{R}.

We need to show that

|x - z| \leq |x - y| + |y - z|

. This follows from the properties of absolute values. Therefore, the set

\mathbb{R}

with the distance function

d(x, y) = |x - y|

forms a metric space. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Elijah Allen on Solvelet

1. Determine whether the set of real numbers

\mathbb{R}

with the absolute value metric is complete.2. Find the open ball

B(x_0, r)

in the metric space

\mathbb{R}^2

with the Euclidean distance metric, where

x_0 = (1, 2)

and

r = 3

DefinitionA metric space is defined in a set containing a metric (i.e., distance function) which defines the distance between two element. These properties include non-negativity, symmetry, triangle inequality and identity of indiscernibles. Ex) Real numbers R with the usual distance ( d(x, y) = | x-y.

Related Calculators

Temperature Calculator Laplace Transform of Integrals Calculator Markov Chains Calculator Express in Terms of Sine and Cosine Calculator

Need topic explanation ? Get video explanation