Gate-2015 Power set

Example:1

If the power set of A is denoted as P(A) and A = \left\{5,\left\{6\right\}, \left\{7\right\}\right\}  Which of the following options are true?

A.    \phi \in 2^{A}

B.    \phi \subseteq 2^{A}

C.    \left\{5,\left\{6\right\}\right\} \in 2^{A}

D.    \left\{5,\left\{6\right\}\right\} \subseteq 2^{A} 

 

  1. A and C only
  2. B and C only
  3. A , B and C only
  4. A, B and D only

 

Solutions:-

 

Answer

 A = \left\{5,\left\{6\right\}, \left\{7\right\}\right\}

P(A) = { ϕ ,  {5} ,   {{6}}    ,  {{7}}  ,  {5, {6}}  ,  {5, {7}}   ,  { {6} , {7} } , {5, {6} , {7}} }

Clearly \phi \in 2^{A} and  \phi \subseteq 2^{A} But  

\left\{5,\left\{6\right\}\right\} \in 2^{A} means the elements { 5, {6}} belongs  to P(A)  

\left\{5,\left\{6\right\}\right\} \subseteq 2^{A} means the elements  { 5, {6}}  is subest of P(A)  which is wrong . Think why??

Because an element of a set can't be the subset of a set.  Instead of { 5, {6}}  if { { 5, {6}} } was given then that could be correct.

Example:

A={1,2,3} 

1 can't be a subset of A but {1} is a subset of A. Before a  set s' become a subset of some set S, s' should be a set first.

 

So only A, B, C are correct.

 

 

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