Question

For a set $A$, consider the following statements:
$\begin{align}
  & 1)A\cup P(A)=P(A) \\
 & 2)\{A\}\cap P(A)=A \\
 & 3)P(A)-\{A\}=P(A) \\
\end{align}$
where $P$ denotes the power set. Which of the statements given above is/are correct?
A. 1 only
B. 2 only
C. 3 only
D. 1, 2, and 3

Answer 1

Hint: In this question, we are to find the correct statement from the given statements about a power set. The power set consists set of all subsets of a set. It is denoted by $P(A)$. Thus, by the definition of a power set, we can define the above-given statements to be true or false.



Formula Used:Set: A collection of objects in which it is possible to decide whether a given object belongs to the collection or not is said to be a set. Those objects are nothing but the elements in the set.
Sets are represented in two ways: The roaster method and the Set builder form
Roaster method:
The set of vowels – \[\left\{ a,\text{ }e,\text{ }i,\text{ }o,\text{ }u \right\}\]
The set of natural numbers – \[\left\{ 1,\text{ }2,\text{ }3,\text{ }\ldots. \right\}\]
Set builder form:
The set of vowels – \[\left\{ x:x\text{ }is\text{ }a\text{ }vowel\text{ }in\text{ }English\text{ }alphabet \right\}\]
The set of natural numbers – \[\left\{ x:x\text{ }is\text{ }a\text{ }natural\text{ }number \right\}\]
Some of the important mathematical sets:
$N$ - the set of natural numbers - $N=\{1,2,3...\}$
$Z$- the set of integers - $Z=\{0,\pm 1,\pm 2,\pm 3,...\}$
Power set: The set of all subsets of a set $A$ is said to be a power set of $A$ and it is denoted by $P(A)$
If $A=\{1,2,3\}$ then its power set is
$P(A)=\{\varnothing ,\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\}\}$
If set $A$ contains $n$ elements, then $P(A)$ contains ${{2}^{n}}$ elements.



Complete step by step solution:The given statements are:
$\begin{align}
  & 1)A\cup P(A)=P(A) \\
 & 2)\{A\}\cap P(A)=A \\
 & 3)P(A)-\{A\}=P(A) \\
\end{align}$
According to the definition of power set i.e., the set of all subsets of a set $A$ is said to be a power set of $A$ and it is denoted by $P(A)$, only statement (1) is true.
Verification:
Consider $A=\{1,2\}$, then $P(A)=\{\varnothing ,\{1\},\{2\},\{1,2\}\}$
$\begin{align}
  & A\cup P(A)=\{1,2\}\cup \{\varnothing ,\{1\},\{2\},\{1,2\}\} \\
 & \text{ }=\{\varnothing ,\{1\},\{2\},\{1,2\}\} \\
 & \Rightarrow A\cup P(A)=P(A) \\
\end{align}$
Thus, (1) is true.



Option ‘A’ is correct

Here, the definition of the power set of a set is used to prove the true statement. Here the set $P(A)\subseteq A$ and $A\subseteq \{A\}$, then $P(A)\subseteq \{A\}$. By using this relation, we can verify the above-given statements.



Note: