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Question:
If $ n(P) = 8,n(Q) = 10 $ and $ n(R) = 5 ('n' $ denotes cardinality) for three disjoint sets $ P, Q, R $ then $ n (P \cup Q \cup P ) = $
If $ n(P) = 8,n(Q) = 10 $ and $ n(R) = 5 ('n' $ denotes cardinality) for three disjoint sets $ P, Q, R $ then $ n (P \cup Q \cup P ) = $
Updated On: Jun 23, 2024
- $ 23 $
- $ 20 $
- $ 18 $
- $ 15 $
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The Correct Option is A
Solution and Explanation
We have, $n \left(P\right) =8$, $n\left(Q\right)=10$ and $n\left(R\right)=5$
Since $P, Q$ and $R$ are disjoint sets
$\therefore n \left(P\cap Q\right)=n \left(Q\cap R\right)=n\left(R\cap P\right)$
$=n\left(P\cap R\cap Q\right)=0$
Now, $n\left(P\cup Q \cup R\right)$
$=n\left(P\right)+n\left(Q\right)+n\left(R\right)$
$-n\left(P\cap Q\right)-n\left(Q\cap R\right)-n \left(R\cap P\right)+n\left(P\cap Q\cap R\right)$
$=8+10+5-0-0-0+0$
$=23$
Since $P, Q$ and $R$ are disjoint sets
$\therefore n \left(P\cap Q\right)=n \left(Q\cap R\right)=n\left(R\cap P\right)$
$=n\left(P\cap R\cap Q\right)=0$
Now, $n\left(P\cup Q \cup R\right)$
$=n\left(P\right)+n\left(Q\right)+n\left(R\right)$
$-n\left(P\cap Q\right)-n\left(Q\cap R\right)-n \left(R\cap P\right)+n\left(P\cap Q\cap R\right)$
$=8+10+5-0-0-0+0$
$=23$
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Concepts Used:
Sets
Set is the collection of well defined objects. Sets are represented by capital letters, eg. A={}. Sets are composed of elements which could be numbers, letters, shapes, etc.
Example of set: Set of vowels A={a,e,i,o,u}
Representation of Sets
There are three basic notation or representation of sets are as follows:
Statement Form: The statement representation describes a statement to show what are the elements of a set.
- For example, Set A is the list of the first five odd numbers.
Roster Form: The form in which elements are listed in set. Elements in the set is seperatrd by comma and enclosed within the curly braces.
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Set Builder Form:
- The set builder representation has a certain rule or a statement that specifically describes the common feature of all the elements of a set.
- The set builder form uses a vertical bar in its representation, with a text describing the character of the elements of the set.
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