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Solutions to homework # 4. 1. The functions d 3 and d 4 are not metrics since they do not satisfy axiom (a); for example, d 3 (1 , - 1) = 0, and d 4 (2 , 1) = 0. The remaining functions all satisfy (a) and (b), so it remains to see whether they satisfy the triangle inequality (c). To see that d 1 fails the triangle inequality, take x = 1, y = 0, z = 1 / 2. Thus d 1 is not a metric. The function d 2 satisﬁes the trianngle inequality, since the condition q | x - y | ≤ q | x - z | + q | z - y | is equivalent to | x - y | ≤ | x - z | + | z - y | + 2 q | x - z | · | z - y | and the latter is satisﬁed due to the triangle inequality for the absolute value function |·| and due to the fact that the term 2 q | x - z | · | z - y | is nonnegative. So, d 2 is a metric. Finally, d 5 also satisﬁes the triangle inequality and is therefore a metric. Indeed, the triangle inequality for d 5 is equivalent (after some algebraic manipulations) to the inequality | x - y | ≤ | x - z | + | z - y | + 2 | x - z | · | z
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This document was uploaded on 11/17/2010.
- Spring '09