Is U(x,y) = max { |x1-y1|, |x2-y2|} a Valid Metric?

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In summary, a metric in science is a standard of measurement used to quantify and compare physical quantities. It must possess certain characteristics, such as objectivity and reproducibility, and have a defined unit of measurement. It cannot be subjective and is important in ensuring standardized and accurate measurements in scientific research. The appropriate metric for a study depends on the research question and the physical quantity being measured.
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math-chick_41
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I need help on how to show something is a metric.
U: R2 x R2 ----> R where U(x,y) = max { |x1-y1|,|x2-y2|}
I can do all the properties except for the dreaded triangle inequality property. help me out please, thanks
 
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What are your ideas so far?
Here's the basic start, let a=|x1-y1|,b=|x_2-y2|,c=|x3-y3|. (a,b and c are three arbitrary non-negative numbers).

To show:
max(a,b)<=max(a,c)+max(b,c)

Looks like, if it isn't obvious, you can always exhaust different cases.
 

FAQ: Is U(x,y) = max { |x1-y1|, |x2-y2|} a Valid Metric?

What is a metric in science?

A metric in science refers to a standard of measurement used to quantify and compare different physical quantities. It is a fundamental tool used to make accurate observations and draw conclusions in various scientific fields.

How do you show something is a metric?

To show that something is a metric, it must possess certain characteristics. It must be objective, reproducible, and have a defined unit of measurement. Additionally, it should be able to measure a wide range of values and be consistent and precise in its measurements.

Can a metric be subjective?

No, a metric cannot be subjective. It must be based on objective and measurable criteria to be considered a valid metric. Subjective measures are based on personal opinions and cannot be standardized or replicated, making them unreliable in scientific research.

Why is it important to use metrics in science?

Using metrics in science allows for standardized and accurate measurements, ensuring that data can be compared and replicated by other scientists. It also helps to eliminate bias and subjectivity, making scientific research more reliable and credible.

How do you choose the appropriate metric for a scientific study?

Choosing the appropriate metric for a scientific study depends on the specific research question and the physical quantity being measured. It is essential to select a metric that is relevant, precise, and can accurately capture the data needed to answer the research question.

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