Answer to Question #94840 in Discrete Mathematics for Amrit

Question #94840
Prove or disprove: If A, B, and C are nonempty sets, and A×B = A×C, then B = C.
1
Expert's answer
2019-09-23T07:12:11-0400

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


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