Discussion
Let A = {2, 3, 1} and B = {4, 5, 6} be subsets of X = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Which of the following describes \(A^{C} \cap B\) ?
Answer: A
\(A^{C} \cap B = \) {4, 5, 6}.
The complement of A within X is everything in X except 1, 2 and 3:
\[ A^{C} = \{0, 4, 5, 6, 7, 8, 9, 10\} \]
Intersecting with B = {4, 5, 6} keeps the elements in both sets, which is all of B, since A and B have no element in common. In general, when A and B are disjoint, \(A^{C} \cap B = B\).
{1, 2, 3, 4, 5} mixes in elements of A, which the complement excludes, and {2, 4, 5} and {5, 6, 8} include elements that are not in B or drop ones that are.
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