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

**Exercises:**

**Question 1:**

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

**Question 2:**

If A is a subset of the set S, prove that :

2a)

2b)

2c)

2d)

**Question 3:**

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.

(b)

(c) .

**Question 4:**

If A, B, C are sets, then prove that :

i)

ii)

iii)

iv)

**Question 5:**

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

**Question 6:**

Prove that if , , and are functions, then

**Question 7:**

Let be a function, let A and B be subsets of X, and let C and D be subsets of Y. Prove that:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

**Question 8:**

(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 .

Cheers,

Nalin Pithwa

PS: These tutorial problems can be used for IIT JEE Maths, Pre RMO, RMO Maths etc. also.

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