pls explain?

Question

for every set S & every metric d on S, which of the following is a metric on S?
1)4+d
2)(e^d)-1
3)d-mod(d)
4)d^2
5)sq root of d

Answer 1

For function to be a metric it has to satisfy
a) D(x, x) >=0
b) D(x, y) = D(y, x)
c) D(x, y) <= D(x, z) + D(z,y)

if d is a metric, then
d(x, x) > = 0 and D(x, x) = d(x, x) + 4 is also > = 0
Also
D(x, y) = d( x, y) + 4 = d(y, x) + 4 = D(y, x).....since d itself is a metric.
and D(x, y) = d(x, y) + 4 < = d( x, z) + d(y, z) + 4 < =
d(x, z) + 4 + d( y, z) + 4 = D(x, z) + D(y,z)
It satisfies all three properties, hence it is also a metric.

Answer 2

4+d

Answer 3

1) all the others are either powers or modulus.