Answer to Question #94840 in Discrete Mathematics for Amrit

Question

Prove or disprove: If A, B, and C are nonempty sets, and A×B = A×C, then B = C.

Answer 1

Let $A$, $B$, 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.)