Reference: Introductory Real Analysis, Kolmogorov and Fomin, Dover Publications.
Problem 1:
Prove that if
and
, then
.
Problem 2:
Show that in general
.
Problem 3:
Let
and
. Find
and
.
Problem 4:
Prove that (a) 
Prove that (b) 
Problem 5:
Prove that 
Problem 6:
Let
be the set of all positive integers divisible by
. Find the sets (i)
(ii)
.
Problem 7:
Find (i)
(ii) 
Problem 8:
Let
be the set of points lying on the curve
where
. What is
?
Problem 9:
Let
for all real x, where
is the fractional part of x. Prove that every closed interval of length 1 has the same image under f. What is the image? Is f one-to-one? What is the pre-image of the interval
? Partition the real line into classes of points with the same image.
Problem 10:
Given a set M, let
be the set of all ordered pairs on the form
with
, and let
if and only if
. Interpret the relation R.
Problem 11:
Give an example of a binary relation which is:
- Reflexive and symmetric, but not transitive.
- Reflexive, but neither symmetric nor transitive.
- Symmetric, but neither reflexive nor transitive.
- Transitive, but neither reflexive nor symmetric.
We will continue later, 🙂 🙂 🙂
PS: The above problem set, in my opinion, will be very useful to candidates appearing for the Chennai Mathematical Institute Entrance Exam also.
Nalin Pithwa