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Different Types of Sets

  1. A set having finite number of elements is called a finite set.
  2. A set which is not a finite set is called an infinite set.
     
    For example, let A = set of all points on a particular straight line. Here A is an infinite set.
  3. A set having no element is called a null set or an empty set or a void set. It is denoted by φ or {}.
  4. A set having single element is called a singleton set, e.g., {φ}, {1}, {3}.
  5. Two finite sets A and B are said to be equivalent if they have the same cardinal number. Thus sets A and B are equivalent iff n(A) = n(B).
  6. Two sets A and B are said to be equal if each element of A is an element of B and each element of B is an element of A.
Notes:
  • Two sets A and B are equal if x ∈ A ⇒ x ∈ B and x ∈ B ⇒ x ∈ A.
  • Equal sets are equivalent sets but equivalent sets may or may not be equal.
  1. Subsets: A set A is said to be a subset of a set B if each element of A is also an element of B. If A is a subset of set B, we write A B. Thus, A B x A x B.
    • Every set is its own subset.
    • Empty set is a subset of each set.
    • Let A and B be any two sets, then A = B A B and B A.
    • Let A, B, C be three sets. If A B and B C, then A C.
  2. A set A is said to be a proper subset of a set B if A is a subset of B and AB.
  3. A set A is said to be a superset of set B, if B is a subset of A, i.e., each element of B is an element of A. If A is a superset of B, we write A B.
  4. The set or family of all the subsets of a given set A is said to be the power set of A and is denoted by P(A). Symbolically, P(A) = {X : X A}. Thus, X P(A)  X A. If A has n elements then its power set P(A) has 2n elements, i.e., A B P(A) P(B).
  5. In any discussion in set theory we need a set such that all sets under consideration in that discussion are its subsets. Such a set is called the universal set for that discussion.




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