If P = {1,2,3,4,5} and P $\cup$ Q = {1,2,3,4,5,6,7}, list the elements in Q

Analysis

The question asks us to find the set Q, given the set P and the union of P and Q ($P \cup Q$).

Concept

The union of two sets ($P \cup Q$) contains all the elements that are in set P, in set Q, or in both. To find the elements of Q, we look for any elements present in the union ($P \cup Q$) that are not already accounted for in set P.

Step-by-Step Solution

1. Identify the given sets: • $P = \{1, 2, 3, 4, 5\}$ • $P \cup Q = \{1, 2, 3, 4, 5, 6, 7\}$

2. Compare the sets: • We see that the elements $\{1, 2, 3, 4, 5\}$ are already in set P. • The elements $\{6, 7\}$ are present in the union ($P \cup Q$) but are missing from set P.

3. Determine Q: • Since $P \cup Q$ must contain all elements from both sets, the elements $\{6, 7\}$ must belong to set Q to complete the union. • Therefore, $Q = \{6, 7\}$.

Conclusion

Comparing our result to the options provided:

• A. {6}
• B. {7}
• C. {6, 7}
• D. {5, 6}

The correct set is {6, 7}, which corresponds to option C.

Exam Tip

When dealing with set operations, think of the union ($\cup$) as a "marriage" of sets. If you have the result of the marriage and one partner, you can easily find the other partner by identifying who is "missing" from the first partner's list but present in the final result!