Question

Show that \(E^{2}\) becomes a metric space if distances \(\rho(\bar{x}, \bar{y})\) are defined by (a) \(\rho(\bar{x}, \bar{y})=\left|x_{1}-y_{1}\right|+\left|x_{2}-y_{2}\right|\) or (b) \(\rho(\bar{x}, \bar{y})=\max \left\\{\left|x_{1}-y_{1}\right|,\left|x_{2}-y_{2}\right|\right\\}\) where \(\bar{x}=\left(x_{1}, x_{2}\right)\) and \(\bar{y}=\left(y_{1}, y_{2}\right) .\) In each case, describe \(G_{\overline{0}}(1)\) and \(S_{\overline{0}}(1) .\) Do the same for the subspace of points with nonnegative coordinates.

Short answer

Both metrics satisfy metric properties making \(E^2\) a metric space, with specific shapes for open and closed balls in each metric.

Step-by-step solution

Check Non-negativity

For both metrics, we need to show the distance \(\rho(\bar{x}, \bar{y})\) is always non-negative. (a) For \(\rho(\bar{x}, \bar{y}) = |x_1 - y_1| + |x_2 - y_2|\), the absolute value ensures that each term is non-negative, so \(\rho(\bar{x}, \bar{y}) \geq 0\).(b) For \(\rho(\bar{x}, \bar{y}) = \max \{|x_1 - y_1|, |x_2 - y_2|\}\), the max of non-negative numbers is also non-negative, hence \(\rho(\bar{x}, \bar{y}) \geq 0\).

Check Identity of Indiscernibles

We need to show \(\rho(\bar{x}, \bar{y}) = 0\) if and only if \(\bar{x} = \bar{y}\).(a) \(\rho(\bar{x}, \bar{y}) = 0\) implies \(|x_1 - y_1| + |x_2 - y_2| = 0\), which implies each absolute term is zero, so \(x_1 = y_1\) and \(x_2 = y_2\), thus \(\bar{x} = \bar{y}\).(b) \(\rho(\bar{x}, \bar{y}) = 0\) implies \(\max {|x_1 - y_1|, |x_2 - y_2|} = 0\), which means both \(|x_1 - y_1| = 0\) and \(|x_2 - y_2| = 0\), thus \(\bar{x} = \bar{y}\).

Check Symmetry

We need \(\rho(\bar{x}, \bar{y}) = \rho(\bar{y}, \bar{x})\).(a) \(\rho(\bar{x}, \bar{y}) = |x_1 - y_1| + |x_2 - y_2| = |y_1 - x_1| + |y_2 - x_2| = \rho(\bar{y}, \bar{x})\).(b) \(\rho(\bar{x}, \bar{y}) = \max\{|x_1 - y_1|, |x_2 - y_2|\} = \max\{|y_1 - x_1|, |y_2 - x_2|\} = \rho(\bar{y}, \bar{x})\).

Check Triangle Inequality

We need \(\rho(\bar{x}, \bar{y}) \leq \rho(\bar{x}, \bar{z}) + \rho(\bar{z}, \bar{y})\) for both metrics.(a) Use the property of absolute values: \(\rho(\bar{x}, \bar{y}) = |x_1 - y_1| + |x_2 - y_2| \leq (|x_1 - z_1| + |z_1 - y_1|) + (|x_2 - z_2| + |z_2 - y_2|) = \rho(\bar{x}, \bar{z}) + \rho(\bar{z}, \bar{y})\).(b) For max metric, \(\rho(\bar{x}, \bar{y}) = \max\{|x_1 - y_1|, |x_2 - y_2|\}\) satisfies \(\max\{a, b\} \leq \max\{a, c\} + \max\{c, b\}\) due to properties of max functions.

Describe Open and Closed Ball with Metric (a)

Using the metric \(\rho(\bar{x}, \bar{y}) = |x_1 - y_1| + |x_2 - y_2|\): - Open ball \(G_{\bar{0}}(1)\) is described by \(|x_1| + |x_2| < 1\), which is a diamond shape.- Closed ball \(S_{\bar{0}}(1)\) is described by \(|x_1| + |x_2| \leq 1\).

Describe Open and Closed Ball with Metric (b)

Using the metric \(\rho(\bar{x}, \bar{y}) = \max\{|x_1 - y_1|, |x_2 - y_2|\}\):- Open ball \(G_{\bar{0}}(1)\) is described by \(\max\{|x_1|, |x_2|\} < 1\), which is a square.- Closed ball \(S_{\bar{0}}(1)\) is described by \(\max\{|x_1|, |x_2|\} \leq 1\).

Subspace with Nonnegative Coordinates

For both metrics, consider only points \(\bar{x} = (x_1, x_2)\) where \(x_1, x_2 \geq 0\). - For metric (a), the sets \(\{|x_1| + |x_2| < 1\}\) and \(\{|x_1| + |x_2| \leq 1\}\) restrict to nonnegative sector of diamond.- For metric (b), the sets \(\{\max\{|x_1|, |x_2|\} < 1\}\) and \(\{\max\{|x_1|, |x_2|\} \leq 1\}\) restrict to nonnegative quadrant of square.

Key concepts

triangle inequality

The Triangle Inequality is a fundamental property in metric spaces. It states that for any three points in a space, the distance between two points is always less than or equal to the sum of the distances from an intermediary point. Let's break it down to see why this makes intuitive sense.

• Imagine you are traveling from point A to point C, with a stop at point B. The direct route from A to C should logically be shorter than the sum of going from A to B and then B to C.
• In mathematical terms for a metric space, if you have three points \( \bar{x} \), \( \bar{y} \), and \( \bar{z} \), the triangle inequality principle is expressed as: \[ \rho(\bar{x}, \bar{y}) \leq \rho(\bar{x}, \bar{z}) + \rho(\bar{z}, \bar{y}) \]

This inequality ensures a kind of logical coherence in the 'shape' and 'feel' of the metric space, making distances consistent and understandable.

open ball

The concept of an Open Ball is quite similar to imagining a bubble with a certain radius in a space. Inside this ball, you have all points that are strictly less than the given radius from a fixed center point.

• When you say \( G_{\bar{0}}(1) \) in a metric space, you are talking about the collection of points whose distance from the origin is less than 1.
• This can be visualized like a circle in two dimensions (or a sphere in three). Everything inside is included, but the edge is not.

In the specific context of the problem, different metrics describe different shapes of open balls:

• For the metric \( \rho(\bar{x}, \bar{y}) = |x_1 - y_1| + |x_2 - y_2| \), the open ball consists of points forming a diamond shape whose sum of coordinates is less than 1.
• With the metric \( \rho(\bar{x}, \bar{y}) = \max\{|x_1 - y_1|, |x_2 - y_2|\} \), the open ball is like a square, having maximum coordinate components less than 1.

closed ball

A Closed Ball in a metric space is similar to its open ball counterpart, but it includes the boundary or the edge of the ball.

• When referring to \( S_{\bar{0}}(1) \), it represents the set of all points, including those exactly on the boundary, whose distance from the origin is less than or equal to 1.
• Again, it's like a filled circle in two dimensions or a filled sphere in three dimensions with its edge also being part of the set.

In our context:

• The metric \( \rho(\bar{x}, \bar{y}) = |x_1 - y_1| + |x_2 - y_2| \) gives us a diamond-shaped closed ball where the sum of absolute values of coordinates is less than or equal to 1.
• For the metric \( \rho(\bar{x}, \bar{y}) = \max\{|x_1 - y_1|, |x_2 - y_2|\} \), it forms a square closed ball where the maximum component is less than or equal to 1.

distance function

The Distance Function is a way of measuring how far two points are from each other in a space. A metric space uses a set of rules or a function to consistently determine this distance.

• For any metric, it's crucial to satisfy four properties: non-negativity, identity of indiscernibles, symmetry, and triangle inequality. These were carefully checked in the problem.
• Non-negativity ensures no negative distances, which makes sense because distance can't be less than zero.
• Identity of indiscernibles means if distance is zero, then both points are identical.
• Symmetry implies it doesn't matter which two points are chosen first; the distance is the same either way.
• Triangle inequality, as earlier discussed, maintains logical coherence and consistency within the space.