Solution to Prove or disprove: If A, B, and C are nonempty sets, and A×B = A×C, … - Sikademy
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Prove or disprove: If A, B, and C are nonempty sets, and A×B = A×C, then B = C.

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Let AB, and C be nonempty sets such that A\times B = A\times C. Since A is nonempty, there is a\in A. Now we prove that B = C. Let x\in B. Then \langle a, x\rangle\in A\times B by the definition of set product. Since A\times B = A\times C\langle a, x\rangle\in A\times C, and x\in C again by the definition of set product. Therefore, B is included in C. By a symmetric proof, C is included in B. Therefore, B = C. (Actually, the assumption that B and C are nonempty is not needed.)

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