Question

For two sets A and B, let AΔB denote the set of elements which belong to A or B but not both. If P = {1,2,3,4}, Q = {2,3,5,6,}, R = {1,3,7,8,9}, S = {2,4,9,10}, then the number of elements in (PΔQ)Δ(RΔS) is

• A. 9
• B. 7
• C. 6
• D. 8

Answer 1

The Correct Option is B

Step 1: Define Sets 

\[ P = \{1, 2, 3, 4\}, \quad Q = \{2, 3, 5, 6\} \]

Step 2: Find \( P \Delta Q \)

The symmetric difference \( P \Delta Q \) is the set of elements which are in either \( P \) or \( Q \) but not in both:

\[ P \Delta Q = \{1, 4, 5, 6\} \]

Step 3: Define Sets R and S

\[ R = \{1, 3, 7, 8, 9\}, \quad S = \{2, 4, 9, 10\} \]

Step 4: Find \( R \Delta S \)

The symmetric difference \( R \Delta S \) is:

\[ R \Delta S = \{1, 2, 3, 4, 7, 8, 10\} \]

Step 5: Now Compute \( (P \Delta Q) \Delta (R \Delta S) \)

We now compute the symmetric difference between the two results:

\[ (P \Delta Q) = \{1, 4, 5, 6\}, \quad (R \Delta S) = \{1, 2, 3, 4, 7, 8, 10\} \]

Now combine and remove common elements (which appear in both sets):\ Common elements: \( 1, 4 \)

\[ \text{Final result: } \{2, 3, 5, 6, 7, 8, 10\} \]

Step 6: Count the Elements

\[ \text{Number of elements} = \boxed{7} \]