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.

Let us examine the following collections:

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

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

● Prime factors of 30 i.e. 2, 3, 5.

● Triangles 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.

● Five most renowned scientists of the world.

● Seven most beautiful girls in a society.

● 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