Answered

Let $P$ be the set of integers that are multiples of 3 between 1 and 20 and $Q$ be the set of even natural numbers up to 15. Which of the following sets is NOT a subset of the union of $P$ and $Q$?


A$\{3,5,6,12\}$
B$\{6,12\}$
C$\{3,6,12\}$
D$\{3,6,12,15\}$

Verified Answer :

  • Define \(P = \{3, 6, 9, 12, 15, 18\}\) and \(Q = \{2, 4, 6, 8, 10, 12, 14\}\).
  • The union \(S = P \cup Q = \{2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 18\}\).
  • Check each option to see if every element is in \(S\).
  • Since \(5\) is not in \(S\), the set \(\{3, 5, 6, 12\}\) is not a subset of \(S\).
  • Final Answer: \(\boxed{\{3, 5, 6, 12\}}\)

Explanation

  1. Defining Sets P and Q

First, define sets \(P\) and \(Q\) based on the given conditions.
\(P\) is the set of multiples of \(3\) between \(1\) and \(20\). These integers are \(3, 6, 9, 12, 15, 18\). Thus, \(P = \{3, 6, 9, 12, 15, 18\}\).
\(Q\) is the set of even natural numbers up to \(15\). These integers are \(2, 4, 6, 8, 10, 12, 14\). Thus, \(Q = \{2, 4, 6, 8, 10, 12, 14\}\).

  1. Finding the Union of P and Q

Next, determine the union of \(P\) and \(Q\), denoted as \(S = P \cup Q\). The union contains all elements that are in \(P\), in \(Q\), or in both.
Combining the elements: \(S = \{2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 18\}\).

  1. Evaluating Candidate Subsets

Now, check which of the provided candidate sets is not a subset of \(S\). A set is a subset of \(S\) if all its elements are contained in \(S\).

  1. For \(\{3, 5, 6, 12\}\): The element \(5\) is not in \(S\). Thus, this set is not a subset of \(S\).
  2. For \(\{6, 12\}\): Both \(6\) and \(12\) are in \(S\). Thus, this set is a subset of \(S\).
  3. For \(\{3, 6, 12\}\): All elements \(3, 6, 12\) are in \(S\). Thus, this set is a subset of \(S\).
  4. For \(\{3, 6, 12, 15\}\): All elements \(3, 6, 12, 15\) are in \(S\). Thus, this set is a subset of \(S\).

Therefore, the set that is not a subset of \(P \cup Q\) is \(\{3, 5, 6, 12\}\).

Examples

This problem helps in understanding set operations such as unions and subset verification. In logistics or inventory management, you might have two different groups of items (e.g., items in storage unit P and items in storage unit Q). Knowing the total collection (the union) allows you to determine if a specific order or shipment (represented as a subset) can be fulfilled using your current stock. If a requested shipment contains an item not present in your combined stock, you know that the order cannot be fulfilled.

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