Verify the identity:
let us observe first that each of the fifth degree expression is just a quadratic in two variables x and y. Let us say the above identity to be verified is:
Use binomial expansion. It is a very longish tedious method.
Factorize each of the quadratic expressions using quadratic formula method (what is known in India as Sridhar Acharya’s method):
Now fill in the above details.
You will conclude very happily that :
The above identity is transformed to :
You will find that and
Hence, it is verified that the given identity . QED.
Prove that a function f is 1-1 iff for all . Given that .
Prove that a function if is onto iff for all . Given that .
(a) How many functions are there from a non-empty set S into \?
(b) How many functions are there from into an arbitrary set ?
(c) Show that the notation implicitly involves the notion of a function.
Let be a function, let , , and . Prove that
Let I be a non-empty set and for each , let be a set. Prove that
(a) for any set B, we have
(b) if each is a subset of a given set S, then where the prime indicates complement.
Let A, B, C be subsets of a set S. Prove the following statements:
🙂 🙂 🙂
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.
Given that and are the roots of the quadratic equation , find the value of
Let and be the two roots of the given quadratic equation:
By Viete’s relations between roots and coefficients:
and but we also know that
Now, let us call which in turn is same as
We have already determined in terms of p and q above.
Now, again note that which in turn gives us that so we get:
Hence, the given expression E becomes:
, which is the desired solution.
🙂 🙂 🙂