It’s a strange way of starting a lecture that I adopt..sometimes…I first give my students quizzes or exams…Here is some foundation mathematics for my deserving students and also, if any of my reader is interested:
Basic Notions from Set Theory:
Reference: Introduction to Analysis, Maxwell Rosenlicht, Dover Publications,
Dover Pub, math link: http://store.doverpublications.com/by-subject-mathematics.html
Let be the set of real numbers and let the symbols have their conventional meanings:
a) Show that
b) List the elements of
c) Show that
If A is a subset of the set S, prove that :
Let A, B, C be elements of a set S. Prove the following statements and illustrate them with diagrams:
(a) … a De Morgan law. In words, it can be said that the union of two complements is the complement of the intersection of the two.
If A, B, C are sets, then prove that :
Let f be a non-empty set and for each , let X be a set. Prove that
(i) for any set B, we have :
(ii) if each is a subset of a given set S, then
Prove that if , , and are functions, then
Let be a function, let A and B be subsets of X, and let C and D be subsets of Y. Prove that:
(a) Prove that a function f is one-to-one if and only if for all .
(b) Prove that a function f is onto if and only if for all .
PS: These tutorial problems can be used for IIT JEE Maths, Pre RMO, RMO Maths etc. also.