Prove or disprove: If A, B, and C are nonempty sets, and A×B = A×C, then B = C.
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Let , , and be nonempty sets such that . Since is nonempty, there is . Now we prove that . Let . Then by the definition of set product. Since , , and again by the definition of set product. Therefore, is included in . By a symmetric proof, is included in . Therefore, . (Actually, the assumption that and are nonempty is not needed.)