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20. If [tex]\( P \)[/tex] is the set of multiples of [tex]\( 6 \)[/tex] less than [tex]\( 50 \)[/tex] and [tex]\( Q \)[/tex] is the set of multiples of [tex]\( 12 \)[/tex] less than [tex]\( 50 \)[/tex], find [tex]\( n(P \cap Q) \)[/tex].



Answer :

GinnyAnswer
To solve the problem, we need to determine the number of elements in the intersection of two sets: \(P\) and \(Q\).

1. Define the sets:
- Set \(P\) is the set of multiples of \(6\) that are less than \(50\).
- Set \(Q\) is the set of multiples of \(12\) that are less than \(50\).

2. List the elements of each set:
- The multiples of \(6\) less than \(50\) are:
\(6, 12, 18, 24, 30, 36, 42, 48\)
Hence, \(P = \{6, 12, 18, 24, 30, 36, 42, 48\}\)
- The multiples of \(12\) less than \(50\) are:
\(12, 24, 36, 48\)
Hence, \(Q = \{12, 24, 36, 48\}\)

3. Find the intersection of \(P\) and \(Q\):
- The intersection of two sets includes only the elements that are present in both sets.
- The common elements in both \(P\) and \(Q\) are:
\(12, 24, 36, 48\)
Therefore, \(P \cap Q = \{12, 24, 36, 48\}\)

4. Count the number of elements in the intersection:
- The number of elements (also known as the cardinality) in the set \(P \cap Q\) is given by counting the elements in \(\{12, 24, 36, 48\}\).
- There are \(4\) elements in the intersection set.

5. Conclusion:
- The number of elements in the intersection of sets \(P\) and \(Q\) is \(4\).

Thus, [tex]\(n(P \cap Q) = 4\)[/tex].

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