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 $(P\Delta Q)\Delta(R\Delta S)$ is

Accepted Answer

A. 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 num

Symmetric difference of sets

Explanation

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