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

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