MA1102 Calculus Tutorial 1

Topic Covered:

Language of sets.

Textbook: Chapter 1

It is important to understand the language that mathematics is written. The fundamental notion of a set underlies all mathematical objects. It is only with a good command of the language then only we can benefit from the ideas being communicated. The language must be universal and that means precision and thoroughness. This is not always achieved. But allow time to reach a certain degree of perfection. You will need a good knowledge of the elements or most oftenly, the properties of the elements for you to be able to use the set for either further investigation or application. You will be looking at various finite and infinite sets constructed out of integers or real numbers and will be asked question about relation between sets (or subsets ) and operations such as union, intersection, complement, relative complement. Some of the properties are described in terms of simple inequality such as that you would have learnt in school. The aim is precision in giving the answers to the questions posed. Be precise to the point. That means know the definition well enough, familiar enough and confident enough to put down your answer unequivocally. This may be hard for some of you and so we have prepared some easier first steps in the form of activities for you to go through. Be patient and try them out. You will soon reap the benefit of knowing this language well. The tutorial proper will ask you slightly deeper questions. Question 6 is an example of application. The notion of Cartesian product is defined as the collection of all 'ordered pairs'. We assume a heuristic definition of 'ordered pair'. We ask for the observation of geometric properties that are capable of extending the question to intervals of the real numbers. We assume the existence of the real numbers here without searching for the reason for its existence.

Recommended Textbook: Ng Tze Beng., Calculus, An Introduction, Springer

References: K.G. Binmore, Sets, Numbers and Logic

J. A. Green, Sets and Groups,

Halmos, Naive Set Theory, (Hard).