Question

If \[U{\text{ }} = {\text{ }}\left\{ {1,2,3} \right\}\]and \[A{\text{ }} = {\text{ }}\left\{ {1,2} \right\}\]then \[\left[ {P\left( A \right)} \right]'{\text{ }} = {\text{ }}..............\]
\[\left( 1 \right)\]\[\left\{ {{\text{ }}\left\{ 3 \right\}{\text{ }},{\text{ }}\left\{ {2{\text{ }},{\text{ }}3} \right\}{\text{ }},{\text{ }}\left\{ {1{\text{ }},{\text{ }}3} \right\}{\text{ }},{\text{ }}\left\{ {1{\text{ }},{\text{ }}2} \right\}{\text{ }},{\text{ }}\phi {\text{ }}} \right\}\]
\[\left( 2 \right)\]\[\;\left\{ {{\text{ }}\left\{ 3 \right\}{\text{ }},{\text{ }}\left\{ {2{\text{ }},{\text{ }}3} \right\}{\text{ }},{\text{ }}\left\{ {1{\text{ }},{\text{ }}3} \right\}{\text{ }},{\text{ }}\left\{ {1{\text{ }},{\text{ }}2{\text{ }},{\text{ }}3} \right\}{\text{ }}} \right\}\]
\[\left( 3 \right)\]\[\left\{ {{\text{ }}\left\{ 3 \right\}{\text{ }},{\text{ }}\left\{ {2{\text{ }},{\text{ }}3} \right\}{\text{ }},{\text{ }}\left\{ {1{\text{ }},{\text{ }}3} \right\}{\text{ }},{\text{ }}\left\{ {1{\text{ }},{\text{ }}2{\text{ }},{\text{ }}3} \right\},{\text{ }}\phi } \right\}\]
\[\left( 4 \right)\]\[\left\{ {{\text{ }}\left\{ 3 \right\}{\text{ }},{\text{ }}\left\{ {2{\text{ }},{\text{ }}3} \right\}{\text{ }},{\text{ }}\left\{ {1{\text{ }},{\text{ }}3} \right\}{\text{ }},{\text{ }}\left\{ {1{\text{ }},{\text{ }}2} \right\}{\text{ }}} \right\}\]

Answer 1

Hint: We have to find the complement of the power set of $A$ . We solve this question by using the concept of power set and complement of power set of $A$ . We find the complement of the power set of $A$then subtract the elements of the power set of $A$ from the elements of the power set of the universal set $U$ . Each element of the power set is included in these brackets \[\left\{ {\text{ }} \right\}\].

Complete step-by-step answer:
The intersection of two terms or events are said to the common elements shared by the two events . The symbol of intersection is also stated as “ and “ . whereas the union of the two terms or events is said to be the sum total of the two events and subtracting the common portion or the intersection of the two events or terms . The symbol of union is also stated as “ or “ .
Given : \[U{\text{ }} = {\text{ }}\left\{ {1{\text{ }},{\text{ }}2{\text{ }},{\text{ }}3} \right\}\]
\[A{\text{ }} = {\text{ }}\left\{ {1{\text{ }},{\text{ }}2} \right\}\]
Subsets of $U$ are \[\left\{ 1 \right\}{\text{ }},{\text{ }}\left\{ 2 \right\}{\text{ }},{\text{ }}\left\{ 3 \right\}{\text{ }},{\text{ }}\left\{ {1{\text{ }},{\text{ }}2} \right\}{\text{ }},{\text{ }}\left\{ {1{\text{ }},{\text{ }}3} \right\}{\text{ }},{\text{ }}\left\{ {2{\text{ }},{\text{ }}3} \right\}{\text{ }},{\text{ }}\left\{ {1{\text{ }},{\text{ }}2{\text{ }},{\text{ }}3} \right\}{\text{ }},{\text{ }}\phi \]
Power set of \[U{\text{ }} = {\text{ }}\left[ {{\text{ }}P\left( U \right){\text{ }}} \right]\]
\[\left[ {{\text{ }}P\left( U \right){\text{ }}} \right]{\text{ }} = {\text{ }}\left\{ {{\text{ }}\left\{ 1 \right\}{\text{ }},{\text{ }}\left\{ 2 \right\}{\text{ }},{\text{ }}\left\{ 3 \right\}{\text{ }},{\text{ }}\left\{ {1{\text{ }},{\text{ }}2} \right\}{\text{ }},{\text{ }}\left\{ {1{\text{ }},{\text{ }}3} \right\}{\text{ }},{\text{ }}\left\{ {2{\text{ }},{\text{ }}3} \right\}{\text{ }},{\text{ }}\left\{ {1{\text{ }},{\text{ }}2{\text{ }},{\text{ }}3} \right\}{\text{ }},{\text{ }}\phi {\text{ }}} \right\}\]
Subsets of $A$ are \[\left\{ 1 \right\}{\text{ }},{\text{ }}\left\{ 2 \right\}{\text{ }},{\text{ }}\left\{ {1{\text{ }},{\text{ }}2} \right\}{\text{ }},{\text{ }}\phi \]
Power set of \[A{\text{ }} = {\text{ }}\left[ {{\text{ }}P\left( A \right){\text{ }}} \right]\]
\[\left[ {{\text{ }}P\left( A \right){\text{ }}} \right]{\text{ }} = {\text{ }}\left\{ {{\text{ }}\left\{ 1 \right\}{\text{ }},{\text{ }}\left\{ 2 \right\}{\text{ }},{\text{ }}\left\{ {1{\text{ }},{\text{ }}2} \right\}{\text{ }},{\text{ }}\phi {\text{ }}} \right\}\]
Complement of power set of $A$= power set of $U$ - power set of $A$
\[\left[ {{\text{ }}P\left( A \right){\text{ }}} \right]{\text{' }} = {\text{ }}\left[ {{\text{ }}P\left( U \right){\text{ }}} \right]{\text{ }} - {\text{ }}\left[ {{\text{ }}P\left( A \right){\text{ }}} \right]\]
\[\left[ {{\text{ }}P\left( A \right){\text{ }}} \right]'\]contains the non common elements of \[\left[ {{\text{ }}P\left( U \right){\text{ }}} \right]\]and\[\left[ {{\text{ }}P\left( A \right){\text{ }}} \right]\].
Hence ,
\[\left[ {{\text{ }}P\left( A \right){\text{ }}} \right]{\text{' }} = {\text{ }}\left\{ {{\text{ }}\left\{ 3 \right\}{\text{ }},{\text{ }}\left\{ {1{\text{ }},{\text{ }}3} \right\}{\text{ }},{\text{ }}\left\{ {2{\text{ }},{\text{ }}3} \right\}{\text{ }},{\text{ }}\left\{ {1{\text{ }},{\text{ }}2{\text{ }},{\text{ }}3} \right\}{\text{ }}} \right\}\]
Thus , The correct option is\[\;\left( 2 \right)\]
So, the correct answer is “Option 2”.

Note: The collection of all the subsets of a set $A$ is called the power set of $A$ . It is denoted by \[P\left( A \right)\]. In general , if $A$ is a set with \[n\left( A \right){\text{ }} = {\text{ }}m\], then it can be shown that $n[P(A)] = {2^m}.$
Complement of a subset $A$ of the universal set $U$ is the set of all elements of $U$ which are not the elements of $A$ which can be calculated using the Venn diagram .