Reference: Introductory Real Analysis, Kolmogorov and Fomin, Dover Publications.
Prove that if and , then .
Show that in general .
Let and . Find and .
Prove that (a)
Prove that (b)
Let be the set of all positive integers divisible by . Find the sets (i) (ii) .
Find (i) (ii)
Let be the set of points lying on the curve where . What is ?
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.
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.
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.