# Set Theory

A set is a collection of well-defined objects. The members of a set can be literally anything like letters of English alphabet, or different types of alphabets, or name of the countries, or numbers, or marks obtained by a student.Given, capital letters are representing a standard set.

A = (a, b, c, d, e, f) â€“ The first six letters of english alphabet

B = (US, China, Japan, India) â€“ The names of the top four countries in terms of their GDP

C = (2, 4, 6, 8, 10) â€“ The first five even natural numbers

Here, A, B and C are different sets which are representing different group of objects.

In this chapter, we will confine ourselves with

- Type of sets
- Solving techniques
- Maxima and minima

Different ways of representing a set:

**Roster Method**With the help of this method, a set is represented by all the elements of it written under the brackets separated by commas.**Set Builder Method**With the help of this method, a set is represented by the common property of all its elements. It is written as*x*|*x*,*p*(*x*) holds}*x*:*x*,*p*(*x*) holds}; where*p*(*x*) is the common property shared by all the elements of set A.*x*e N |*x*â‰¤ 5), which can be written with the roster method as A = {1, 2, 3, 4, 5}

# Types of Sets

A set having no element is known as a empty or a null set and it is denoted as Ï† or { }.*Empty or Null Set*A set having only one element is known as a singleton set.*Singleton set*A set having a countable number of elements is known as the finite set.*Finite set*A set whose elements cannot be counted is known as infinite set.*Infinite set*Two sets are said to be equal sets if all the elements of set A are included in set B and all the elements of set B are included in set A. If two sets A and B are equal then it is represented by A = B and if A and B are not equal then it is written as AÂ¹ B, that is, all the elements of set A are not included in set B and all the elements of set B are not included in set A.*Equal sets*A is said to be the subset of another set B if all the elements of set A are included in set B. â€˜Set A is subset of Set Bâ€™ is shown by A âŠ† B. We can now say that every element of set A is a member of Set B.*Subsets Set*- Every set is a subset of itself.
- Every set has an empty set as its subset.
- Total number of subsets of a set having
*n*elements in 2.^{n}

A set which contains all the sets in a given context is a universal set.*Universal Set**N*is the universal set.The collection of all the subsets of a set is known as the power set of that set. If A is the set, then a set containing all the subsets of A is known as the power set of A. It is denoted by P(A).*Power Set*^{2}and the subsets are {}, {1}, {2} and {1, 2} and the set containing all these four sets is known as the power set represented as P(A).Swiss mathematician Euler first gave the idea of representing sets by a diagram. Later on, British mathematician Venn brought this into practice. So, it is known as Eulerâ€“Venn diagram or simply Venn diagram. In this way of representing sets we use a closed curve, generally a circle, to denote sets and their operations.*Venn Diagram*