Symmetric difference of sets

Recall that symmetric difference $\Delta$ is associative and commutative, and $x \in X_1 \Delta X_2 \Delta \dots \Delta X_k$ if and only if $x$ belongs to an odd number of the sets $X_i$.

Let's evaluate each element in $P, Q, R, S$:

• 1: belongs to $P, R$ (2 sets - even) $\implies$ not included
• 2: belongs to $P, Q, S$ (3 sets - odd) $\implies$ included
• 3: belongs to $P, Q, R$ (3 sets - odd) $\implies$ included
• 4: belongs to $P, S$ (2 sets - even) $\implies$ not included
• 5: belongs to $Q$ (1 set - odd) $\implies$ included
• 6: belongs to $Q$ (1 set - odd) $\implies$ included
• 7: belongs to $R$ (1 set - odd) $\implies$ included
• 8: belongs to $R$ (1 set - odd) $\implies$ included
• 9: belongs to $R, S$ (2 sets - even) $\implies$ not included
• 10: belongs to $S$ (1 set - odd) $\implies$ included

The elements appearing an odd number of times are $\{2, 3, 5, 6, 7, 8, 10\}$. The number of elements is 7.