Question

Example B={do,re,mi,fa,so}
What are the subsets of b
What are the proper subsets of b

Answer 1

No. of subsets of B = 32
No. of proper subsets = 31
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Here ,
No. of Subsets of B are
$\binom{5}{0} + \binom{5}{1} + \binom{5}{2} + \binom{5}{3} + \binom{5}{4} + \binom{5}{5} = {2}^{5} = 32$

1) Selecting none out of 5 :
That is,
$\binom{5}{0} = 1$
Subset is : null set or
{¤}

2) Selecting 1 out of 5 :
$\binom{5}{1} = 5$
Subsets are :
{do} , {re}, {mi} , {fa} , {so}

3) Selecting 2 out of 5:
$\binom{5}{2} = 10$
Subsets are :
{do, re} , {do, mi} ,{do, fa} ,{do, so} , {re, mi} , {re, fa} , {re, so} ,{mi, fa},{mi, so}, {fa, so}.

4) Selecting 3 out of 5 :
$\binom{5}{3} = 10$
Subsets are :
{do, re, mi} , {do, re, fa} , {do, re,so} , { do, mi, fa} , {do, mi, so} ,{do, fa, so},{re, mi, fa} ,{re,mi,so} ,{mi, fa, so}, & {re, fa, so} .

5) Selecting 4 out of 5 :
$\binom{5}{4} = 4$
Subsets are :
{do, re, mi, fa} ,{do, we, mi, so} , {re, mi , fa, so} ,
{mi, fa, so, do} ,{ fa, so, do, re}

6) Selecting 5 out of 5:
$\binom{5}{5} = 1$

{do, re, mi, fa, so}


=> These above all are total 32 subsets of A.
=> Out of them 1 is improper which 5 out of 5 .
that is {do, re, mi, fa, so}.
=> Remaining all 31 are proper subsets of A.

Answer 2

Formula:

The number of subsets of a set having m elements is

2^m

Solution:

The subsets of B are

{ }

{do} , {re}, {mi} , {fa} , {so} 

{do, re} , {do, mi} ,{do, fa} ,{do, so} , {re, mi} , {re, fa} , {re, so} ,{mi, fa},{mi, so}, {fa, so}.

{do, re, mi} , {do, re, fa} , {do, re,so} ,
{ do, mi, fa} , {do, mi, so} ,{do, fa, so},
{re, mi, fa} ,{re,mi,so} ,{mi, fa, so},
{re, fa, so} .

{do, re, mi, fa} , {do, we, mi, so} ,
{re, mi , fa, so} ,
{mi, fa, so, do} ,{ fa, so, do, re} 

{do, re, mi, fa, so} 


No. of subset of B is 2^5 = 32

No. of proper subset of B is 31.