A student is told that AUB is the set of all elements that belong to \( A \) or to \( B \). However, when asked to find the union of \( A=\{a, b, c, d\} \) and \( B=\{c, d, e, f, g, h\} \), the student says that the answer is \( \{a, b, e, f, g, h\} \). How is the word "or" in the definition of set union being misunderstood? Give an everyday example where "or" is used in the mathematical sense. Choose the correct answer below. Use the Show Work feature to provide related calculations and additional reasoning to support your answer. A. The student interpreted AUB as AnB, and listed the correct set of elements in AnB. B. The student interpreted "or" as "and" in which both possibilities hold. C. The student interpreted AUB as BUA, and listed the correct set of elements in BUA. D. The student interpreted "or" as the "exclusive or" in which either one of two possibilities holds, but not both.
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The misunderstanding stems from the interpretation of "or." The student mistakenly viewed "or" as "exclusive or," meaning they thought that the elements could either belong to set \( A \) or set \( B \), but not both at the same time. In this case, they excluded the common elements \( c \) and \( d \) from the union. In everyday language, "or" is often used in a way that implies one choice but not both, such as when someone says, "You can have tea or coffee," suggesting you must choose one but not both. To clarify the correct interpretation, the union \( A \cup B \) should include all elements from both sets, leading to the correct result of \( \{a, b, c, d, e, f, g, h\} \). So, the right answer is: D. The student interpreted "or" as the "exclusive or" in which either one of two possibilities holds, but not both.
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