State whether the following statement is true or false. If the statement is false,rewrite it correctly.
If $A$ and $B$ are non-empty sets,then $A \times B$ is a non-empty set of ordered pairs $(x, y)$ such that $x \in A$ and $y \in B$.
If $A$ and $B$ are non-empty sets,then $A \times B$ is a non-empty set of ordered pairs $(x, y)$ such that $x \in A$ and $y \in B$.
(A) The statement is True.
By the definition of the Cartesian product of two sets $A$ and $B$,$A \times B = \{(x, y) : x \in A \text{ and } y \in B\}$. If $A$ and $B$ are non-empty,there exists at least one element $a \in A$ and $b \in B$,which implies $(a, b) \in A \times B$. Thus,$A \times B$ is a non-empty set.
By the definition of the Cartesian product of two sets $A$ and $B$,$A \times B = \{(x, y) : x \in A \text{ and } y \in B\}$. If $A$ and $B$ are non-empty,there exists at least one element $a \in A$ and $b \in B$,which implies $(a, b) \in A \times B$. Thus,$A \times B$ is a non-empty set.
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