☼ Subsets  :   Let A and B be two sets given in such a way that every element of  A is in B, then we say that A is a subset of  B and we write, A  B.

Results  On  Subsets  :  

 (i)  is  a subset of  every set.

 (ii) Every  set  is  a  subset  of   itself.

 (iii) The number of  all possible subsets of  a set containing  n  elements is  2ⁿ.

 (iv) The number of  all proper subsets of  a set containing n elements is  ( 2ⁿ – 1 ).

 (v) The set of  all subsets of  a given set A is called the power set of  A, denoted by  P ( A ).   If  A  has  n  elements,  P ( A ) has  2ⁿ  elements.

 e.g. (i) Write down the power set of  A = { a, b, c}

Soln. Subsets of  A =  , {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, A

P ( A )  =  { , {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, A }

  (ii) Write down the power set of  A = { a, { b, c}}

  Soln. Let   A = { a, B },  where,  B  =  { b, c}

Subsets of  A =   , {a}, B, A

P ( A )  =  {   , {a}, B, A }  =  {   , {a}, {{b, c}}, A }.

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