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

x

y
 ≤

x

z

+

z

y

is equivalent to

x

y
 ≤ 
x

z

+

z

y

+ 2

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

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

y

+

x

y
 · 
x

z
 · 
z

y

,
which holds because of the usual triangle inequality for
 · 
.
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 Spring '09
 Topology, Decimal, Metric space

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