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Q. Prove that |d(x,z)-d(y,z)|≤d(x,y)

Q. Show that another metric D on a set S defined by

d(x,y) = ∫_a^b▒〖| x(t)-y(t) |dt 〗

d(x,y) = ∫_a^b▒〖| x(t)-y(t) |dt 〗

Q. If A is a subspace of l^∞ consisting of all sequence of 0 and 1. What is the induced metric on A?

Q.Show that d(x,y)=Sup|∑_j-n_j |, j∈N satisfy triangular property.

Q.Let D be a metric on X determined all constant k s.t

(i) kd is a metric

(ii)(k+d) is a metric

(Prove it)

(i) kd is a metric

(ii)(k+d) is a metric

(Prove it)

Q.Find all metric on a set X consisting of one point and consisting of two points.

Is the real line a metric space?

Q. Which of the following sets are countable or uncountable.

(i) set of real numbers

(ii) set of complex numbers

(i) set of real numbers

(ii) set of complex numbers

Q. Which of the following sets are countable or uncountable.

(i)set of negative integers

(ii) set of rational numbers

(i)set of negative integers

(ii) set of rational numbers

Q. Which of the following sets are countable or uncountable.

(i) set of natural numbers

(ii) set of whole numbers

(iii) set of positive integers

(i) set of natural numbers

(ii) set of whole numbers

(iii) set of positive integers