Question:

If n(P)=8,n(Q)=10 n(P) = 8,n(Q) = 10 and n(R)=5(n n(R) = 5 ('n' denotes cardinality) for three disjoint sets P,Q,R P, Q, R then n(PQP)= n (P \cup Q \cup P ) =

Updated On: Jun 23, 2024
  • 23 23
  • 20 20
  • 18 18
  • 15 15
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The Correct Option is A

Solution and Explanation

We have, n(P)=8n \left(P\right) =8, n(Q)=10n\left(Q\right)=10 and n(R)=5n\left(R\right)=5
Since P,QP, Q and RR are disjoint sets
n(PQ)=n(QR)=n(RP)\therefore n \left(P\cap Q\right)=n \left(Q\cap R\right)=n\left(R\cap P\right)
=n(PRQ)=0=n\left(P\cap R\cap Q\right)=0
Now, n(PQR)n\left(P\cup Q \cup R\right)
=n(P)+n(Q)+n(R)=n\left(P\right)+n\left(Q\right)+n\left(R\right)
n(PQ)n(QR)n(RP)+n(PQR)-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+5000+0=8+10+5-0-0-0+0
=23=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.

  • For example represent the set of vowels in roster form.

A={a,e,i,o,u}

Set Builder Form: 

  1. The set builder representation has a certain rule or a statement that specifically describes the common feature of all the elements of a set.
  2. The set builder form uses a vertical bar in its representation, with a text describing the character of the elements of the set.
  3. For example, A = { k | k is an even number, k ≤ 20}. The statement says, all the elements of set A are even numbers that are less than or equal to 20.
  4. Sometimes a ":" is used in the place of the "|".