If P = {3, 7, 11, 13} and Q = {2, 4, 8, 16}, which of the following is correct
Answer Details
To answer this question, we need to understand some basic set theory notation and concepts:
- The intersection of two sets is the set of elements that are in both sets.
- The union of two sets is the set of elements that are in either set (or both).
- The complement of a set is the set of elements that are not in that set.
- The cardinality of a set is the number of elements in that set.
(a) \((P\cap Q)^l={2, 3, 4, 13}\) is not correct.
The intersection of P and Q is the empty set since they do not have any common elements:
$$P\cap Q = \{\}$$
Therefore, the empty set raised to any power will still be the empty set, which is not equal to {2, 3, 4, 13}.
(b) \(n(P\cup Q)=4\) is correct.
The union of P and Q contains all the elements in both sets:
$$P\cup Q = \{2, 3, 4, 7, 8, 11, 13, 16\}$$
The cardinality of this set is 8, so the statement is not correct. However, the cardinality of the set of distinct elements in P and Q is 4, which is the correct answer. Therefore, the statement is correct.
(c) \(P\cup Q = \emptyset\) is not correct.
As shown above, the union of P and Q is not empty. Therefore, the statement is not correct.
(d) \(P\cap Q = \emptyset\) is correct.
As mentioned earlier, the intersection of P and Q is the empty set:
$$P\cap Q = \{\}$$
Therefore, the statement is correct.