## A nice Set Theory problem

Given sets $A$ and $B$.

Consider the sets $A' = A \times \{0,1\}$ and $B' = B \times \{0,1\}$. That is, $A'$ is the set of all ordered pairs $(a,\varepsilon)$ where $a \in A$ and $\varepsilon \in \{0,1\}$. Similarly for $B'$.

(We deduce that the cardinality, or ‘size’, of $A'$ is two times that of $A$.)

It is known that there exists a bijection from $A'$ to $B'$. Prove that there exists a bijection from $A$ to $B$.

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