text: PROPER SUBSET OF A SET
  
  About "Proper subset of a set"
  
  Proper subset of a set :
  
  A set X is said to be a proper subset of set Y if X ⊆ Y and X ≠ Y.
  
  In symbol, we write X ⊂ Y
  
  Read X ⊂ Y as "X is proper subset of Y"
  
  The figure given below illustrates this.
  
  Proper subset of a set - Examples
  
  Example 1 :
  
  Let A  =  {1, 2, 3, 4, 5} and B  =  { 5, 3, 4, 2, 1}. Determine whether B is a proper subset of A.
  
  Solution :
  
  If B is the proper subset of A, every element of B must also be an element of A and also B must not be equal to A.
  
  In the given sets A and B, every element of B is also an element of A. But B is equal A.
  
  Hence, B is the subset of A, but not a proper subset.
  
  Example 2 :
  
  Let A  =  {1, 2, 3, 4, 5} and B  =  {1, 2, 5}. Determine whether B is a proper subset of A.
  
  Solution :
  
  If B is the proper subset of A, every element of B must also be an element of A and also B must not be equal to A.
  
  In the given sets A and B, every element of B is also an element of A.
  
  And also But B is not equal to A.
  
  Hence, B is a proper subset of A.
  
  Example 3 :
  
  Let A  =  {1, 2, 3, 4, 5} find the number of proper subsets of A.
  
  Solution :
  
  Let the given set contains "n" number of elements.
  
  Then, the formula to find number of proper subsets is
  
  =  2ⁿ¹
  
  The value of "n" for the given set  A is "5".
  
  Because the set A =  {1, 2, 3, 4, 5} contains "5" elements.
  
  Number of proper subsets  =  2¹
  
  =  2
  
  =  16
  
  Hence, the number of proper subsets of A is 16.
  
  Apart from the stuff "Proper subset of a set", let us know some other important stuff about subsets of a set.
  
  Subset of a given set
  
  A set X is a subset of set Y if every element of X is also an element of Y.
  
  In symbol we write
  
  x ⊆ y
  
  Read ⊆ as "X is a subset of Y" or "X is contained in Y"
  
  Read  as "X is a not subset of Y" or "X is not contained in Y"
  
  Power set
  
  The set of all subsets of A is said to be the power set of the set A.
  
  The power set of A is denoted by P(A)
  
  Super set
  
  A set X is said to be a proper subset of set Y if X ⊆ Y and X ≠ Y.
  
  In symbol, we write X ⊂ Y
  
  Here,
  
  Y is called super set of X
  
  Formula to find number of subsets
  
  If A is the given set and it contains "n" number of elements, we can use the following formula to find the number of subsets.
  
  Number of subsets =  2
  
  Formula to find the number of proper subsets :
  
  Number of proper subsets =  2ⁿ¹
  
  Cardinality of power set
  
  We already know that the set of all subsets of A is said to be the power set of the set A and it is denoted by P(A).
  
  If A contains "n" number of elements, then the formula for cardinality of power set of A is
  
  n[P(A)]  =  2
  
  Note :
  
  Cardinality of power set of A and the number of subsets of A are same.
  
  Null set is a subset or proper subset
  
  Null set is a proper subset for any set which contains at least one element.
  
  For example, let us consider the set A  =  { 1 }
  
  It has two subsets. They are { } and { 1 }.
  
  Here null set is proper subset of A. Because null set is not equal to A.
  
  If null set is a super set
  
  If null set is a super set, then it has only one subset. That is { }.
  
  More clearly, null set is the only subset to itself. But it is not a proper subset.
  
  Because, { }  =  { }
  
  Therefore, A set which contains only one subset is called null set.
  
  After having gone through the stuff given above, we hope that the students would have understood "Proper subset of a set".
  
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