Is the Euclidean Metric on RxR and C a Valid Metric?

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SUMMARY

The discussion focuses on the validity of the Euclidean metric on the sets R (real numbers) and C (complex numbers). It confirms that the Euclidean metric on R x R satisfies the three necessary conditions of a metric: non-negativity, identity of indiscernibles, and symmetry. Additionally, it establishes that the Euclidean metric on C x C also qualifies as a metric. Furthermore, the discussion explores the generalization of the Euclidean metric to n-tuples of real numbers (R^n) and complex numbers (C^n), affirming that the metric properties hold in these cases as well.

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  • Familiarity with Euclidean geometry concepts
  • Basic knowledge of real and complex number systems
  • Ability to work with n-tuples in mathematical contexts
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Mathematicians, students of advanced calculus, and anyone studying metric spaces and their properties will benefit from this discussion.

fabbi007
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R is real line, C is set of Complex numbers

If we considered the Euclidean metric on RXR

a. Show whether the Euclidean metric on R
RXR is a metric.
b. Show whether the Euclidean metric on C
C is a metric.
c. Generalize the Euclidean metric to a set made up of all n-tuples of real numbers
X=R power n. Is this also true on C power n?

Can I get some direction on solving this proofs?
 
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a. Show whether the Euclidean metric on RxR is a metric.

There is a definition for "metric". Check all 3 clauses of that definition in this particular case.
 

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