Set theory

Set theory

The set theory was developed by George Cantor in 1845-1918. Today, it is used in almost every branch of mathematics and serves as a fundamental part of present-day mathematics.

In set theory we will learn about representation in roster form and set builder form , types of sets (Empty set, singleton set, finite and infinite sets, equal and equivalent sets), cardinal number of a set, subsets (Proper subset, super set, power set), number of proper subsets, universal set, operation on sets (Union, intersection, difference and complement of sets).

In everyday life, we often talk of the collection of objects such as a bunch of keys, flock of birds, pack of cards, etc. In mathematics, we come across collections like natural numbers, whole numbers, prime and composite numbers. 

(Ref Britannia. Math)

Let us examine the following collections:

1 Even natural numbers less than 20, i.e., 2, 4, 6, 8, 10, 12, 14, 16, 18. 

2Vowels in the English alphabet, i.e., a, e, i, o, u. 

3 Prime factors of 30 i.e. 2, 3, 5. 

4Triangles on the basis of sides, i.e., equilateral, isosceles and scalene. 

We observe that these examples are well-defined collections of objects.

Let us examine some more collections. 

1 five most renowned scientists of the world. 

2 Seven most beautiful girls in a society. 

3 Three best surgeons in America. 

These examples are not well-defined collections of objects because the criterion for determining as most renowned, most beautiful, best, varies from person to person. 

Sets:

A set is a well-defined collection of distinct objects.

We assume that,

● The word set is synonymous with the word collection, aggregate, class and comprises of elements.

● Objects, elements and members of a set are synonymous terms.

● Sets are usually denoted by capital letters A, B, C, ....., etc.

● Elements of the set are represented by small letters a, b, c, ....., etc.

If ‘a’ is an element of set A, then we say that ‘a’ belongs to A. We denote the phrase ‘belongs to’ by the Greek symbol ‘∈‘ (epsilon). Thus, we say that a ∈ A.

If ‘b’ is an element which does not belong to A, we represent this as b ∉ A.

Some important sets used in mathematics are

N: the set of all natural numbers = {1, 2, 3, 4, .....}

Z: the set of all integers = {....., -3, -2, -1, 0, 1, 2, 3, .....}

Q: the set of all rational numbers

R: the set of all real numbers

Z+: the set of all positive integers

W: the set of all whole numbers

In other words

1. the branch of mathematics which deals with the formal properties of sets as units (without regard to the nature of their individual constituents) and the expression of other branches of mathematics in terms of sets.

Concept of set theory:- 

set theory, branch of mathematics that deals with the properties of well-defined collections of objects, which may or may not be of a mathematical nature, such as numbers or functions. The theory is less valuable in direct application to ordinary experience than as a basis for precise and adaptable terminology for the defination of complex and sophisticated mathematical concepts.

Between the years 1874 and 1897, the German mathematician and logician Georg Cantor created a theory of abstract sets of entities and made it into a mathematical discipline. This theory grew out of his investigations of some concrete problems regarding certain types of infinite sets of real numbers. A set wrote Cantor, is a collection of definite, distinguishable objects of perception or thought conceived as a whole. The objects are called elements  or members of the set.

The theory had the revolutionary aspect of treating infinite sets as mathematical objects that are on an equal footing with those that can be constructed in a finite number of steps. Since antiquity, a majority of mathematicians had carefully avoided the introduction into their arguments of the actual infinite (i.e., of sets containing an infinite of objects conceived as existing simultaneously, at least in thought). Since this attitude persisted until almost the end of the 19th century, Cantor’s work was the subject of much criticism to the effect that it dealt with fictions—indeed, that it encroached on the domain of philosophers and violated the principles of religion. Once applications to analysis began to be found, however, attitudes began to change, and by the 1890s Cantor’s ideas and results were gaining acceptance. By 1900, set theory was recognized as a distinct branch of mathematics.

At just that time, however, several contradictions in so-called naive set theory were discovered. In order to eliminate such problems, an axiomatic basis was developed for the theory of sets analogous to that developed for elementary geometry. The degree of success that has been achieved in this development, as well as the present stature of set theory, has been well expressed in the Nicolas baurbakis (begun 1939; “Elements of Mathematics”): “Nowadays it is known to be possible, logically speaking, to derive practically the whole of known mathematics from a single source, The Theory of Sets.”

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