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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) – The first six letters of english alphabet
B = (US, China, Japan, India) – The names of the top four countries in terms of their GDP
C = (2, 4, 6, 8, 10) – 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:
  1. 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.
     
    For example, A = {1, 2, 3, 4, 5}
  2. 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
     
    A = {xxp(x) holds}
     
    A = (x:xp(x) holds}; where p(x) is the common property shared by all the elements of set A.
     
    For example, A = {x e N |x ≤ 5), which can be written with the roster method as A = {1, 2, 3, 4, 5}

Types of Sets

  1. Empty or Null Set A set having no element is known as a empty or a null set and it is denoted as φ or { }.
     
    For example, A = set of even prime numbers excluding 2.
  2. Singleton set A set having only one element is known as a singleton set.
     
    For example, A = set of even prime number/s.
  3. Finite set A set having a countable number of elements is known as the finite set.
     
    For example, A = set of odd numbers from 100 to 890.
  4. Infinite set A set whose elements cannot be counted is known as infinite set.
     
    For example, A = set of all irrational numbers between 2 and 3.
  5. Equal sets 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 A¹ 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.
     
    For example, A = {a, b, c} and B = {c, b, a} are equal sets. Hence, in this case we can write Set A = Set B or simply A = B.
  6. Subsets Set A is said to be the subset of another set B if all the elements of set A are included in set B. ‘Set A is subset of Set B’ is shown by A ⊆ B. We can now say that every element of set A is a member of Set B.
     
    For example, If A = {a, b, c} and B = {a,b,c,d,e} then A ⊆ B, or A is a subset of B.
     
    Some important results on subsets
    • 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 2n.
  7. Universal Set A set which contains all the sets in a given context is a universal set.
     
    For example, when we are using sets containing natural numbers, N is the universal set.
     
    If A = {a, b, c,}, B = {b, c, d}, C = {c, d}
     
    Then we can take u = {a, b, c, d} as universal set.
     
    Description: 17-1.tif
  8. Power 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).
     
    Let A = {1, 2}, then the number of subsets of this set will be 22 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).
  9. Venn Diagram 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 Euler–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.




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