Set Theory

A set is an unordered collection of different elements. Sets are represented as a collection of well-defined objects or elements and it does not change from person to person.
  • N – Set of all-natural numbers
  • Z – Set of an integer number
  • Q – Set of all rational number
  • R – Set of all real number
  • Z+ - Positive integers
  • W – Whole number
Representation of Sets: -
  • Roster or tabular form: - In this form of representation we list all the elements of the set within braces {} and separate them by commas. Example: - the set of natural numbers less than 5
  • Set Builder form: - In this form of representation we list the properties fulfilled by all the elements of the set. Example: - A = {2, 4, 6, 8}
Types of Set: -
  • Empty Set: - A set that does not contain any element is called an empty set or void set or NULL set. It is denoted by {} or Ø. A set of apples in the basket of groups is an example of an empty set because in a group's basket there are no apples present.
  • Singleton set: - A set that contains a single element is called a singleton set.
    • Example: - There is only one apple in a basket group.
  • Finite Set: - A Set that consists of a definite number of elements are called a finite set.
    • Example: - A set of natural numbers up to 10
  • Infinite Set: - A set that is not finite is called an infinite set. A set in which the process of counting elements does not come to a­^n end is called a^n infinite set.
    • Example: - A set of all-natural numbers.
  • Equivalent set: - If the number of elements is the same for two different sets, then they are called equivalent sets.
  • Equal Set: - The two sets A and B are said to be equal if they have exactly the same elements, the order of elements does not matter.
    • Example: - 

  • Disjoint Set: - The two sets A and B are said to be disjoint if the set does not contain any common element.
    • Example: - 
  • Sub Sets: - A set ‘A’ is said to be a subset of B if every element of A is also an element of B, denoted as A ⊆ B Even the null set is considered to be the subset of another set. In general, a subset is a part of another set.
    • Example: - 
  • Proper Subset: - If A ⊆ B and A ≠ B then A is called the proper subset of b and it can be written as A ⊂ B.
  • Super Set: - If set A is a subset of set B and all the elements of set B are the elements of set A, then A is a superset of set B. It is denoted by A ⊃ B.
  • Universal Set: - A set that contains all the sets relevant to a certain condition is called the universal set. It is the set of all possible values.
Operations on Sets: -
  • Union of Sets: - If set A and set B are two sets, the A union B is the set that contains all the elements of set A and set B. It is denoted as A ∪ B 
    • Example: - 
  • Intersection of Sets: - If set A and set B are two sets then an intersection B is the set that contains only the common elements between set A and set B. It is denoted as A ∩ B. 
    • Example: - 
  • Complement of Sets: - The complement of any set, say P, is the set of all elements in the universal set that are not in set P. It is denoted by P’
  • Difference of Sets: - If set A and set B are two sets then set A difference set B is a set that has elements of A but no elements of B. It is denoted as A – B. 
    • Example: - 
Properties of Sets: -
  • Commutative Properties: -
  • Associative Properties: -
  • Distributive Properties: -
  • Idempotent Law and Law of a NULL and Universal Set: -
  • Properties of the Complement: -
  • Involution Properties: -
  • De Morgan’s Laws: -
  • Universal set – Identity Law: -
Location: Punjab, India

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