Chapter 1. Preliminaries

In this chapter, we review some basic concepts required for understanding the contents of this book.

Sets, Relations, and Functions

Sets

A set is a collection of well-defined objects. Usually the elements of a set have common properties. For example, all the students who enroll for a course ‘Computability’ make up a set. Formally,

Note that the definition of a set is intuitive in nature and was stated by the German mathematician Cantor in 1895. The objects in a set are called the elements or members of the set.

Example 1.1. 

The set E of even positive integers less than 20 can be expressed by:

E = {2, 4, 6, 8, 10, 12, 14, 16, 18}

or

E = {x|x is even and 0 < x < 20}

A set ...

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