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Power Set

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Let A be a non-empty set, then collection of all possible subsets of set A is known as power set. It is denoted by P (A). 
e.g., Suppose A = {1, 2, 3} 

So, P(A) = [$\phi$, {1}, {2}, {3}, {1, 2}, {2, 3}, {3, 1}, {1, 2, 3}]. 

(a) A $\notin$ P(A) 
(b) {A} $\notin$ P(A)

Properties of Power Set 
  1. Each element of a power set is a set. 
  2. If A $\subseteq$ B, then P(A) $\subseteq$ P(B) 
  3. Power set of any set is always non-empty. 
  4. If set A has n elements, then P(A) has $2^n$ elements. 
  5. P(A) $\cap$ P(B) = P(A $\cap$ B) 
  6. P(A) $\cup$ P(B) $\subseteq$ P(A $\cup$ B) 
  7. P(A $\cup$ B) $\neq$ P(A) $\cup$ P(B)  
Sample Problem 1
If set A = {1, 3, 5}, then number of elements in P{P(A)} is
(a) 8
(b) 256
(c) 248
(d) 250 

Answer: (b)
Given, A = {1, 3, 5}
n{P(A)} = $2^3$ = 8, so,
n[P{P(A)}] = $2^8 = 256$

Sample Problem 2
Consider A B = = {1, 2}, {2, 3}. Then which of the following option is correct?
(a) P(A $\cup$ B) $\neq$ P(A) $\cup$ P(B) 
(b) P(A $\cup$ B) = P(A) $\cap$ P(B) 
(c) P(A $\cup$ B) = P(A) $\cup$ P(B) 
(d) None of these

Answer: (a)
Here
P(A) = {$\phi$, {1}, {2}, {1, 2}}
P(B) = {$\phi$, {2}, {3}, {2, 3}}

AB = {1, 2, 3} 
P(A $cup$ B) = {$\phi$, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, {1, 2, 3}}
P(A) P(B) = {$\phi$, {1}, {2}, {3}, {1, 2}, {2, 3}}

So, P(A $\cup$ B) $\neq$ P(A) $\cup$ P(B) 

Sample Problem 3
If A and B are non-empty sets, then P(A) $\cup$ P(B) is equal to
(a) P(A $\cup$ B)
(b) P(A) = P(B) 
(c) P(A $\cap$ B) 
(d) None of these

Answer:
Let A = {1}, B = {2, 3}
A $\cup$ B = {1, 2, 3} and A $\cap$ B = {$\phi$}

Now, P(A) = {$\phi$, {1}}, P(B) = {$\phi$, {2}, {3}, {2, 3}}
P(A) $\cup$ P(B) = {$\phi$, {1}, {2}, {3}, {2, 3}}
P(A $\cup$ B) = {$\phi$, {1}, {2}, {3}, {1, 2}, {2, 3}, {3, 1}, {1, 2, 3}} and
P(A $\cap$ B) = {$\phi$} 

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